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Zo of two wire open line
PMG, Course of Technical Instruction (old) says that the 276Log(2D/d) is
inaccurate when the clearance between the wires is small compared with the wire diameter. The book goes on to say that the acosh formula is an accurate one for all spacings. No mention is made of efffects of skin depth et al on Zo. Alan Owen Duffy wrote: I have found two common expressions for the Zo of a two wire line in space. .............. This suggests (states) that the acosh expression (fully) accounts for proximity effect on Zo. Comments? Owen |
#12
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Zo of two wire open line
Roy Lewallen wrote in
: 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. Hi Roy, Thanks again for the research. I would have thought that current would distribute itself in the conductors for the least voltage drop along the conductor due to inductance and bulk resistance. That suggests that as the conductors are brought very close together, one expects that Zo should decrease smoothly(ie the Zo is a monotonically increasing funtion of distance between conductors for all positive distance). If I take acosh(be/(2*ae)), it has a minimum around be/(2*ae)=1.3. In the expression ae= a * sqrt(1 - (2a/b)^2), ae goes to infinity as b approaches 2a. This would drive acosh(be/(2*ae)) towards infinity as the wires are bought together. 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. I think there are probably limits for use of these formula. The turning point in the region of (2*ae)=1.3 concerns me. I cannot intuitively see an explanation for such a behaviour. Owen |
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