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Zo of two wire open line
I have found two common expressions for the Zo of a two wire line in space. One expression is Zo=276log(D/d). The second is Zo=120acosh(D/d). I have been searching for information on whether the acosh expression takes into account proximity effect. Because it does not consider conductivity or permeability of the conductors, one wonders if it does. Laport has a graph that shows the log expression and a proximity corrected line which turns out to be a very close fit to the acosh line over the range that he plots. The curves are compared in http://www.vk1od.net/balun/Ruthroff/R07.png . I have another reference, Marchand (1947), which gives the log expression then, the acosh expresssion and says "The hyperbolic cosine is obtained because the currents are attracted to one another and become more and more confined to the inside surface as the wires are brought closer together." This suggests (states) that the acosh expression (fully) accounts for proximity effect on Zo. Comments? Owen |
#2
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Zo of two wire open line
"Owen Duffy" wrote in message ... I have found two common expressions for the Zo of a two wire line in space. One expression is Zo=276log(D/d). The second is Zo=120acosh(D/d). I have been searching for information on whether the acosh expression takes into account proximity effect. Because it does not consider conductivity or permeability of the conductors, one wonders if it does. Laport has a graph that shows the log expression and a proximity corrected line which turns out to be a very close fit to the acosh line over the range that he plots. The curves are compared in http://www.vk1od.net/balun/Ruthroff/R07.png . I have another reference, Marchand (1947), which gives the log expression then, the acosh expresssion and says "The hyperbolic cosine is obtained because the currents are attracted to one another and become more and more confined to the inside surface as the wires are brought closer together." This suggests (states) that the acosh expression (fully) accounts for proximity effect on Zo. Comments? Owen I must be doing something wrong. 276log(2)=83 but looks like about 150 on your graph. No? John |
#3
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Zo of two wire open line
"John KD5YI" wrote in
news:Z2srj.1036$qw4.677@trnddc02: I must be doing something wrong. 276log(2)=83 but looks like about 150 on your graph. No? Sorry, that expression should be 276log(2D/d) for a consistent meaning of D and d. (Some people use wire radius and some, diameter... so D here is centre to centre distance, and d is diameter.) Owen |
#4
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Zo of two wire open line
On Sun, 10 Feb 2008 00:08:33 GMT, Owen Duffy wrote:
I have found two common expressions for the Zo of a two wire line in space. One expression is Zo=276log(D/d). The second is Zo=120acosh(D/d). Hi Owen, I have the following expressions found in "Reference Data for Radio Engineers," 22-22 Transmission Lines: Zo = 120 acosh(D/d) Zo ~ 276 log(2D/d) Zo ~ 120 Ln(2D/d) where the symbol ~ means "approximately equal to." 73's Richard Clark, KB7QHC |
#5
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Zo of two wire open line
Richard Clark wrote in
: On Sun, 10 Feb 2008 00:08:33 GMT, Owen Duffy wrote: I have found two common expressions for the Zo of a two wire line in space. One expression is Zo=276log(D/d). The second is Zo=120acosh(D/d). Hi Owen, I have the following expressions found in "Reference Data for Radio Engineers," 22-22 Transmission Lines: Zo = 120 acosh(D/d) Zo ~ 276 log(2D/d) Zo ~ 120 Ln(2D/d) where the symbol ~ means "approximately equal to." Thanks Richard. Yes, I saw those in that publication. ln(x) is a good approximation of acosh(x) for large x, so that explains one approximation, and the log expression is just a scaling of the ln expression. I put some importance on the difference between the approximately equal and unqualified equal signs. If the acosh expression fully accounts for proximity effect, it is interesting that it is independent of frequency, conductivity, and permeability. It would be nice to find a derivation. I think if you derive C and L from first principles, you don't get the same curve as acosh... which suggests that acosh is account for something that L and C from first principles (assuming uniform current distribution) doesn't. Owen |
#6
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Zo of two wire open line
The acosh formula includes the effect of skin effect on inductance and
capacitance(*), while the simplified log formula is equivalent to considering the currents concentrated along lines at the conductor centers. Neither includes proximity effect. Derivations of the acosh formula can be found in Johnson, _Transmission Lines and Networks_, Sec. 3.8; Chipman, _Transmission Lines_ (Schaum's Outline Series), Sec. 6.6; and other references. I'm sure that a formula taking proximity effect into account would be much more complicated, judging by approximate formulas I've seen for calculating resistance change due to proximity effect. From Chipman (p. 114), about the lack of published coverage of proximity effect: "Fortunately, the combination of circumstances that would require accurate information about th eproximity effect factor for distributed internal inductance occurs rather rarely in transmission line practice. The most unfavorable situation would be a parallel wire line with solid circular conductors, the facing suraces of the conductors being separated by only a few percent of a conductor radius, operating at a frequency to have a/[delta] have a value near 2. These conditions make the distributed internal inductance comparable in magnitude to the distributed external inductance, with a proximity factor that might be as small as 0.8 or 0.85. There is no recognized basis for making an accurate analysis of the total distributed inductance of a line for such a case. .. . . When the facing conductor surfaces are at least a conductor diameter apart (s/2a = 2), the distributed internal inductance will be less than 20% of the total distributed inductance, and the proximity effect factor will me not less that 0.87. . . Proximity effect can then not modify the total distributed inductance value by more than about 2%, and the factor need be known only very roughly. . ." (*)And even this is a high-frequency approximation which assumes that the conductors are at least several skin depths thick. Expressions for line inductance without this assumption involve Bessel functions, which I assume would also appear in expressions for Z0. Roy Lewallen, W7EL Owen Duffy wrote: I have found two common expressions for the Zo of a two wire line in space. One expression is Zo=276log(D/d). The second is Zo=120acosh(D/d). I have been searching for information on whether the acosh expression takes into account proximity effect. Because it does not consider conductivity or permeability of the conductors, one wonders if it does. Laport has a graph that shows the log expression and a proximity corrected line which turns out to be a very close fit to the acosh line over the range that he plots. The curves are compared in http://www.vk1od.net/balun/Ruthroff/R07.png . I have another reference, Marchand (1947), which gives the log expression then, the acosh expresssion and says "The hyperbolic cosine is obtained because the currents are attracted to one another and become more and more confined to the inside surface as the wires are brought closer together." This suggests (states) that the acosh expression (fully) accounts for proximity effect on Zo. Comments? Owen |
#7
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Zo of two wire open line
Sorry, I clicked Send before fixing up my typos in the Chipman
quotation. Here's how it's really written: "Fortunately, the combination of circumstances that would require accurate information about th proximity effect factor for distributed internal inductance occurs rather rarely in transmission line practice. The most unfavorable situation would be a parallel wire line with solid circular conductors, the facing surfaces of the conductors being separated by only a few percent of a conductor radius, operating at a frequency to have a/[delta] have a value near 2. These conditions make the distributed internal inductance comparable in magnitude to the distributed external inductance, with a proximity factor that might be as small as 0.8 or 0.85. There is no recognized basis for making an accurate analysis of the total distributed inductance of a line for such a case. .. . . When the facing conductor surfaces are at least a conductor diameter apart (s/2a = 2), the distributed internal inductance will be less than 20% of the total distributed inductance, and the proximity effect factor will me not less that 0.87. . . Proximity effect can then not modify the total distributed inductance value by more than about 2%, and the factor need be known only very roughly. . ." (*)And even this is a high-frequency approximation which assumes that the conductors are at least several skin depths thick. Expressions for line inductance without this assumption involve Bessel functions, which I assume would also appear in expressions for Z0. Roy Lewallen, W7EL |
#8
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Zo of two wire open line
Roy Lewallen wrote in
: Thanks Roy. I was suspicious that proximity correction could ignore conductivity, permeability, frequency and the actual diameter and distance. Chipman paints a picture that it is pretty messy mathematically, and the flow into Zo would be really messy. What I do glean is that for D/d2, the error is small. That is a whole lot better than the log formula which is poor for D/d10. I have two immediate applications, one is a model for a 1:1 choke balun (http://www.vk1od.net/balun/Guanella/G.1-1.htm) and the other is yet another transmission line loss calculator, along the lines of my existing calculator that is aware of about 100 standard transmission line types, but this one allowing specification of an arbitrary two wire line (http://www.vk1od.net/tl/twllc.htm). For the balun cases, I am very interested in D/d2. I think I will continue to use the acosh expression, and let it calculate without limit on D/d, but qualify the results with a note that proximity effect is not included in Zo, Zo is underestimated for D/d2 and the error may be significant. Roy, thanks for your time to research and type the notes up. Owen |
#9
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Zo of two wire open line
This might be useful:
King, in _Electromagnetic Engineering_, Vol. 1, says (pp. 467-8): ". . .the problem may be analyzed for wires sufficiently far apart and so small in radius that rotational symmetry in the interior and on the surface of each conductor is a good approximation, and then generalized to closely spaced conductors of large radius simply by writing ae= a * sqrt(1 - (2a/b)^2) be = b/2 * (1 + sqrt(1 - (2a/b)^2)) instead of a and b in formulas for the external impedance." a is the wire radius, b the center-center spacing, ae is the radius to be substituted and b is the spacing to be substituted. He's saying that you can account for proximity effect just by substituting equivalent wires of smaller diameter and closer spacing which have idealized evenly distributed current around the outside. He introduces the equation for ae earlier without proof, but references a 1921 paper in an obscure publication, as a way to adjust internal inductance for the proximity effect. The method is much simpler than the approximate equations for added loss due to proximity effect I've come across, so I suspect it's an approximation, but I don't know the applicability limits. But you might try incorporating it and see how it compares with measurements and with calculations based on the more complicated equations. I see that when the wires are just touching (b = 2a), you end up substituting wires of radius zero, spaced at half the actual distance. I suppose that's reasonable. Roy Lewallen, W7EL |
#10
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Zo of two wire open line
Hello Owen,
In electromagnetic waves & antennas (Orfanidis, paragraph 9.5), they calculate the electrostatic capacitance of a two wire conductor assuming the wires as equipotential surfaces (so no E-field components parallel to the conductor surface). This implies non-uniform charge distribution for close distance (so proximity effect). Also when you look in the graph, the charge centers are closer to each other then the center to center distance of the parallel wires. From the capacitance you can directly calculate the characteristic impedance based on c0 propagation speed. The exact solution (based on the capacitor approach) comes with the Acosh function. So I think that (some or all) proximity effect is taken into account. Off course they assume all charge on the surface, so skin depth diameter of wire. The formula with the "ln" is just an approximation for the "acoh" function when D/d is large. The acosh formula gives zero for touching wires. This is correct for the assumption that all charge is on the surface (so no magnetic field will penetrate the wires). Hope this will help you a bit. Best regards, Wim PA3DJS www.tetech.nl (Dutch) Owen Duffy ha escrito: I have found two common expressions for the Zo of a two wire line in space. One expression is Zo=276log(D/d). The second is Zo=120acosh(D/d). I have been searching for information on whether the acosh expression takes into account proximity effect. Because it does not consider conductivity or permeability of the conductors, one wonders if it does. Laport has a graph that shows the log expression and a proximity corrected line which turns out to be a very close fit to the acosh line over the range that he plots. The curves are compared in http://www.vk1od.net/balun/Ruthroff/R07.png . I have another reference, Marchand (1947), which gives the log expression then, the acosh expresssion and says "The hyperbolic cosine is obtained because the currents are attracted to one another and become more and more confined to the inside surface as the wires are brought closer together." This suggests (states) that the acosh expression (fully) accounts for proximity effect on Zo. Comments? Owen |
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