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repair section D_11
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Short revision note by Phil dated 9.9.14, replacing part (b) of Section D.11 in his transmission line appendices. It derives β'^2 = β^2 - k^2 at low ω, using the low-frequency limits of k from Appendix Q, for G>0 (β'^2 ≈ RdcGdc) and G=0. It gives a Belden 8281 coaxial cable example showing β'a << 1, and says parts (c) and (d) need no change.
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Repair Section D.11 PhL 9.9.14
This small repair to D.11 (b) only was installed today 9.9.14.
I will start with section (b) and come back later to Section (a).
(b) Low frequency values for β'
For low ω and G > 0: We seek an expression for β' at low ω. Since β'2 = β2 - k2, we need to know about β and k. For any ω, we know first that
β2 = -jωμσ (1.5.1d) for good conductor
so β2 → 0 as ω→0. Meanwhile, the ultra-low frequency limit of k is known from (Q.4.6) to be,
Re(k) ≈ (ω/2) (Rdc + ωdLdc) + O(ω2) ω < ωd = (σd/εd)
Im(k) ≈ - [ 1 + (tanL/2) (ω/ωd)] + O(ω2) . (Q.4.6)
As ω→0, we find Re(k)→ 0 and Im(k)→ - = -. Thus we have at low ω,
β2 ≈ 0
k ≈ - j => k2 = - RdcGdc
β'2 = β2 - k2 = -k2 ≈ RdcGdc . (D.11.1)
For any reasonable transmission line RdcGdc will very small so the Bessel argument xa = β'a << 1.
Low Frequency Example:
For Belden 8281 coaxial cable, Appendix R below Fig R.6 gives Rdc = .036 and Gdc = 0.338 x 10-14 so β'2 ≈ RdcGdc ≈ 1.22 x 10-16 and then β' ≈ 1.1 x 10-8 m-1. The central conductor has a = 3.94 x 10-4 m so β'a ≈ 5 x 10-12 << 1.
Since β'a is very small, we shall need to evaluate fm, gm and hm for small xa = β'a.
For low ω and G= 0: When Gdc = 0, the ultra-low ω behavior of k is given by (Q.4.9),
Re(k) ≈ + ( 1 - tanL/2) + O(ω3/2)
Im(k) ≈ - ( 1 + tanL/2) + O(ω3/2) . (Q.4.9)
As ω→ 0 we then have
k ≈ ( 1 - j )
k2 = [ RdcC/2] ω [ -2j ] = -jω RdcC
β2 = -jωμσ (1.5.1d) for good conductor
β'2 = β2 - k2 ≈ -jωμσ +jωRdcC = -jω(μσ - RdcC) . (D.11.2)
Again β' is very small at low frequency and in fact β' → 0 as ω→0. Thus again we have β'a << 1 so we shall need to evaluate fm, gm and hm for small xa = β'a.
(c) Low frequency evaluation of fm, gm and hm
No changes needed for this section (c).
(d) Low frequency E fields
I think the rest of this section is all OK. The anomaly remains!