Better expression for R and Li v3

Better expression for R and Li PhL 9.6.14 Here I develop the plots which now appear in App Q section "how good is the crude model". The Maple code for these plots is in "Plots of Li and R vs w Belden.mws " After two major screw-up versions, I am back again to this question. Remember from lines doc (4.11.34) that. R = Re(Zs1+ Zs2) L = Le + (1/ω) Im(Zs1+ Zs2) = Le + Li and these are the R and Li I am trying to make a better model for! For the round wire, I know from Chapter 2 that, Zs(ω) = (2.4.6) at least for the central conductor of a coax cable! So how does this give Rdc in the low ω limit? We can go back a little more to this point from (2.2.30), Ez(r) = (-jω/β) (2.2.29) Ez(a) = (-jω/β) (2.2.27) The small ω limit of the above would involve J0 = 1 J1 = (z/2) Then Ez(r) = (-jω/β) = (-jω/β) = = = = => Jz = = quite reasonable Meanwhile, in the same limit Zs(ω) = = = = = = = Rdc and there is your answer! In Chapter 2 I plot Zz versus δ instead of ω. I could make new plots versus ω and just see what they look like. But basically, here is my model for the center conductor, R1(ω) = Re [ ] ωLi1(ω) = Im [ ] where a = a1 What about the other conductor of Belden cable? I guess I have never addressed this question. I could apply the above formula to widely spaced twin lead. For the Belden, App R shows that the DC Li of the shield is about 1/8th that of the center conductor. .8e-8/.5e-7 = 8/50 = 0.16 = 1/8. App R also shows that at high ω, both R and Li shield/center have a ratio of 1/a2 / 1/a1 = a1/a2 = 1/6.4 ≈ 1/6. So in both limits, Li and R of the shield can be roughly neglected compared to the center conductor. How would this be affected by changing the shield thickness? The hi ω results would be the same, since only the inside surface of the shield matters. But if we make t of the shield larger, we have Li(shield) = [ (4/3)(t/a2) ] = [ (4/3) ] so if we were to fatten the shield, its Li does in fact increase as its R2DC decreases and t increases. I could do a Chapter 2 style solution for the shield and I probably could get some kind of Bessel function formula for its Zs at all ω. But maybe I will just assume we can neglect it more or less, or that I am dealing with widely spaced twin lead. I have done enough detail work I think. So, here are my new plots for the center of Belden 8281. R1(ω) = Re [ ] ω Li1(ω) = Im [ ] where a = a1 OK, here are some good plots: This one shows how Li remains constant at μ/8π out to about 500KHz and then drops off in the expected 1/fashion. It is not so different from my Heaviside plot! Here is a comparison of the above plot to my original Heaviside model, Question: Why is the DC value higher in the Heaviside model? Answer: because I include the effect of the shield. Here is the same plot and a corresponding one for R: Apart from the issue just noted, the Heaviside model is not too bad! Here are the same plots if I just boost the single-wire result by 1.15 to account for the shield. These plots I think show just how good the Heaviside model really is at least for the Belden cable case. This is now incorporated in Appendix Q.