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old tail to section D.9 (d) REVIEWED

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Word document dated 10.6.14 (PhL) holding an old, short tail of Section D.9(d) of the Transmission Lines Appendix D, kept after it was replaced by a much longer version. It gives the modified charge pumping boundary conditions for a mildly conducting dielectric and the rule Nm → (ξd/εd)Nm. It then shows the current becoming I', restates the Ez, Er, Eθ field solutions, and concludes that the effect is to replace C by a complex capacitance C'.

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Old tail to Section D.9 (d) PhL 10.6.14 This tail is very short, and gets replaced with a much longer tail! Just saving off the old here. Above are the "modified" charge pumping boundary conditions which replace (D.2.24) and (D.2.25) for a mildly conducting dielectric, Er(r=a,θ) = (jω/σ) n(θ) (D.2.24) Er(r=a,m) = (jω/σ) Nm . (D.2.25) How then does σd ≠ 0 alter the E field results summarized in box (D.2.33)? The rule is this: Nm → (ξd/εd) Nm everywhere . (D.9.23) For example, for the total current we have I = 2πaN0(ω/k) → I' = 2πaN0(ω/k) (ξd/εd) . (D.9.24) If we now denote by I the actual current in the transmission line (I for G = 0, or I' for G>0) , then the E field solutions as expressed in (D.2.33) do not change form, and we may write Second summary of the E field solutions : Rdc = β'2 = β2 - k2 (D.2.33) conducting dielectric Ez(r,m) = (1/4) ηm I Rdc (aβ') fm fm = [ - ] x = β'r Er(r,m) = (j/4) ηm I Rdc (ak) gm gm = [ + ] xa = β'a Eθ(r,m) = (1/4) ηm I Rdc (ak) hm hm = [ - ] From (D.1.8) we know from (D.2.31c) that I = CV (ω/k), so the effect of turning on G is, I = CV (ω/k) → I' = CV (ω/k) (ξd/εd) = C'V (ω/k) (D.9.25) where C' is the complex capacitance per length as in (1.5.19). Thus, the entire effect of a conducting dielectric results in C → C' so that I = CV (ω/k) → C'V (ω/k) in the field expressions above.