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Word document dated 3.26.05 holding saved material for Appendix Q on k and Z0 math. It summarizes the complex wavenumber k(ω) for a transmission line with R, L, G, C, giving its real and imaginary parts and limiting forms for large ω, small ω with G>0, and small ω with G=0 (Q.5-Q.7). A comment explains that at high frequency the internal inductance vanishes by the skin effect, so the phase velocity approaches the speed of light in the dielectric. Many equation symbols are lost in extraction.
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Appendix Q save stuff PhL 3.26.05
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Summary of results:
Summary of k and its limits (Q.8)
k ≡ -j= -j // definition of the wavenumber k(ω)
k = - j jk = = = + j
a ≡ (R2+ω2L2)1/4 (G2+ω2C2)1/4 = |k|
c ≡ RG - ω2LC
Large ω: Re(k) = ω + // = 1/vd and ω = ω/vd = βd0
Im(k) = - + ω >> (R/L),(G/C) (Q.5)
Small ω, G>0: Re(k) = (ω/2)
Im(k) = - - (ω2/8) ω << (R/L),(G/C) (Q.6)
Small ω, G=0: Re(k) = ω1/2 + ω3/2
Im(k) = - ω1/2 + ω3/2 ω << (R/L) (Q.7)
Comment: In principle, all four quantities R,C,L,G can be functions of ω. According to the analysis of Chapter 4 leading to summary box (4.11.34), if we stay away from the low frequency range, only parameter L shows a strong frequency dependence since L = Le + Li(ω). As shown in the comments below (4.11.34), limω→∞ Li(ω) = 0. This is due to the skin effect and result (C.6.8). Thus, we know that limω→∞ L(ω) = Le. But (4.11.34) says LeC = μdεd = 1/vd2 where vd is the speed of light in the dielectric. Therefore, limω→∞ L(ω)C = LeC = 1/vd2 and thus for large ω we get Re(k) = ω = ω/vd and the phase velocity of the wave equals the dielectric speed of light. As ω drops from the high end, the identification of with 1/vd becomes less accurate.