A lossy coaxial feeder

PHYSICS · Subjective · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Electromagnetic waves and their propagation · Electromagnetic waves and their propagation

Inside the Instrument, Ch Transmission lines, grounding and the bench

Problem

A lossy coaxial feeder, driven at f=10 MHzf = 10\,\mathrm{MHz}, has per-unit-length parameters R′=0.20 Ω m−1R' = 0.20\,\Omega\,\mathrm{m}^{-1}, L′=0.25 μH m−1L' = 0.25\,\mu\mathrm{H\,m^{-1}}, G′=1.0×10−6 S m−1G' = 1.0\times10^{-6}\,\mathrm{S\,m^{-1}} and C′=100 pF m−1C' = 100\,\mathrm{pF\,m^{-1}}. (a) Starting from the telegrapher's equations ∂v∂z=−(R′+L′∂t)i\dfrac{\partial v}{\partial z} = -(R' + L'\partial_t)i and ∂i∂z=−(G′+C′∂t)v\dfrac{\partial i}{\partial z} = -(G' + C'\partial_t)v, substitute a sinusoidal phasor solution and derive the general expressions γ=(R′+jωL′)(G′+jωC′)\gamma = \sqrt{(R' + j\omega L')(G' + j\omega C')} for the propagation constant and Z0=(R′+jωL′)/(G′+jωC′)Z_0 = \sqrt{(R' + j\omega L')/(G' + j\omega C')} for the characteristic impedance. (b) Verify that R′≪ωL′R' \ll \omega L' and G′≪ωC′G' \ll \omega C' at this frequency, and hence use the low-loss expansions, α≈R′/(2Z0)+G′Z0/2\alpha \approx R'/(2Z_0) + G'Z_0/2 and β≈ωL′C′\beta \approx \omega\sqrt{L'C'} to first order, with the leading-order value Z0≈L′/C′Z_0 \approx \sqrt{L'/C'}, to evaluate γ=α+jβ\gamma = \alpha + j\beta and Z0Z_0 numerically.

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