old averaging repair for Sec 4_11

Old averaging repair from Section 4.11 (rejected) 6.6.14 Our theory now has an inconsistency which needs repair. We know that for a general transmission line operating at ω > 0, the current density Jz inside the conductors will not be uniformly distributed. It will be larger in the conductor region closest to the other conductor. This "proximity effect" is discussed in Appendix P from an eddy current point of view, see Fig P.13 for an example. The Jz current non-uniformity can be very dramatic as for example in a transmission line having this cross section, where Jz will be large near the gap and small far from the gap: Fig 4.12 Since Jz is non-uniform in each conductor, so is Ez, and so we expect Ez(x1) to be a strong function of the point x1 on the perimeter of C1, certainly in the above cross section example. This means that the left side of (4.11.6a) is a function of x1 = (x1,y1,z) and x2 = (x2,y2,z) whereas the right side in our theory is a function only of z. To remedy this inconsistency, we now have to think of V and W as having very slight dependence on x1 and x2 which we generally ignore, but which we must face up to in (4.11.6a). In reality we have V(x1,x2) and W(x1,x2). This is a manifestation of the fact that in reality φ ≈ constant and Az ≈ constant on the boundaries (with ≈ and not = ). In the extreme skin effect regime (think a very good conductor), the left side of (4.11.6a) can be a violent function of x1 and x2 as in the case of the above figure, but the left side is always very small, even where it is largest, and its variation can be accommodated by the right side of (4.11.6a) which is the difference of large-valued functions which vary only slightly with x1 and x2. We now proceed in our development of the transmission line equations with this understanding of the functions V and W. As shown in (2.4.1), Ez(x1) can be related to the total current in the conductor i(z) by a quantity known as the surface impedance Zs, so Ez(x1) = Zs1(x1) i1(z) Ez2(x2) = Zs2(x2) i2(z) . (4.11.7) The surface impedance of a perfect conductor is zero. Since i1(z) = -i2(z) = i(z), we rewrite (4.11.6) as, [Zs1(x1) + Zs2(x2)] i(z) = - ∂zV - jωW (4.11.8a) ∂zW = - j (β2/ωV . (4.11.8b) We just noted that Ez(x1) may vary violently around the perimeter of conductor C1, especially for a cross section like that shown above. This of course means that Zs1(x1) also varies violently on the perimeter, causing equation (4.11.8a) to have the same inconsistency as (4.11.6a). We now perform a certain slight of hand. We back up to the first equation of (4.11.4) evaluated at x = x1, Ez(x1) = - ∂zφ12(x1) - jωAz12(x1) or Zs1(x1) i1(z) = - ∂zφ12(x1) - jωAz12(x1) . (4.11.9) Recall now (4.7.3), Jz1(x,y,z) = b1(x,y) i1(z) . (4.7.3) We shall now average both sides of (4.11.9) over the perimeter of C1 weighting by Jz1 at the surface, which we intuitively regard as a measure of the relative significance of a piece of the C1 boundary. Thus, define Zs1 ≡ = = < Zs1(x1)>C1 φ12(z) ≡ = < φ12(x1)>C1 (4.11.10a) and similarly for Az12(z). We do a similar average over the C2 perimeter to obtain Zs2, φ12(z) and Az12(z), Zs2 ≡ = < Zs2(x2)>C2 φ12(z) ≡ = < φ12(x2)>C2 (4.11.10b) We could then define the "active perimeter" distances p1 and p2 for the two conductors as p1 ≡ ∫C1 ds1 Zs1(x1) Jz1(x1) The averaged (4.11.9) now becomes, Zs1 i1(z) = - ∂zφ12(z) - jωAz12(z) . (4.11.11) We then slightly redefine V and W as being the difference of the current-averaged potentials, V(z) ≡ φ12(z) - φ12(z) = < φ12(x1)>C1 - < φ12(x2)>C2 ≡ <V(x1,x2)> W(z) ≡ Az12(z) - Az12(z) = < Az12(x1)>C1 - < Az12(x2)>C2 ≡ <W(x1,x2)> . (4.11.12) Performing the same weighted average on (4.11.8b), we arrive at this adjusted version of (4.11.8), [Zs1 + Zs2] i(z) = - ∂zV(z) - jωW(z) ∂zW(z) = - j (β2/ωV(z) . (4.11.13) where now Zs1 and Zs2 are the constants just defined and V(z) and W(z) are given by (4.11.12). Comment: For transmission lines made of "thin wires", which means thin relative to their spacing, Jz is close to uniform over each conductor and the above averaging process is unnecessary. Recall now (4.11.1) which says W(z) = Le i(z). We reinterpret this equation first as W(x1,x2) = Le(x1,x2) i(z), then we current-average both sides to get W(z) = Le i(z) where Le = <Le(x1,x2>. Then replacing W(z) twice by Le i(z) in (4.11.13) gives, (Zs1 + Zs2) i(z) = - ∂zV(z) - jω Le i(z) Le ∂z i(z) = - j (β2/ωV(z) which we then rearrange as ∂zV(z) = - [ Zs1+ Zs1+ jωLe] i(z) ∂z i(z) = - [ jβ2/(ωLe)] V(z) . (4.11.14) These are the classical transmission line equations. They are usually written in this form: = - z i(z) = - y V(z) with z = R + jωL y = G +jωC . (4.11.15)