telegraph equations INSTALLED

This is the Title PhL 3.26.05 This has been installed into lines doc. (f) Time domain equations The results above are all stated in the frequency domain, but it is a simple matter to convert them to the time domain using jω ↔ ∂t. One then makes these replacements z = R+jωL → L∂t + R y = G+jωC → C∂t + G zy = (R+jωL)( G+jωC) → (R + L∂t)( G + C∂t) = LC∂t2 + (LG+RC)∂t + RG (4.11.38) Here then are selected equations and their translations to the time domain: Transmission Line Equations (4.11.14b) : [ coupled first order PDE's] ∂zV = - z i ∂zV(z,t) = - L∂ti(z,t) - LR i(z,t) ∂z i = -y V ∂z i(z,t) = - C∂tV(z,t) - CGV(z,t) (4.11.39) Transmission Line Wave Equations (4.11.15) [ damped wave equations ] ( ∂z2 - zy) V(z) = 0 [ ∂z2 - LC ∂t2 - (LG+RC)∂t - RG] V(z,t) = 0 ( ∂z2 - zy) i(z) = 0 [ ∂z2 - LC ∂t2 - (LG+RC)∂t - RG] i(z,t) = 0 (4.11.40) If we set the loss factors R and G both to 0 these equations become ∂zV = -L∂ti [ ∂z2 - LC ∂t2] V(z,t) = 0 [ undamped wave equations] ∂z i = -C∂tV [ ∂z2 - LC ∂t2] i(z,t) = 0 // lossless (4.11.41) At large ω one has L ≈ Le (note 1 above) and since (4.11.28) says LeC = με = 1/v2 we conclude that the factor LC appearing in the above wave equations is 1/v2 where v is the dielectric wave velocity. The various transmission line equations shown above in the time domain are often referred to as the telegraph (telegrapher, telegrapher's) equations.