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329lect39 - 39 Lossy lines Lossless TLs we have been...

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39 Lossy lines Δ z C z z + Δ z Δ z L + - Δ z C z z + Δ z V ( z ) I ( z ) Δ z L + - I ( z + Δ z ) Δ z R Δ z G IDEAL LOSSLESS T.L.: REALISTIC LOSSY T.L.: I ( z + Δ z ) I ( z ) V ( z ) Lossless TL’s we have been studying so far are idealizations of real TL’s which are invariably lossy . Here, we are making reference to Ohmic energy losses in the conducting wires of the TL, as well as to losses in the imperfect dielectric separating the two conductors. The e ff ect of wire losses in TL’s is modelled by adding a Δ z R resis- tance in series with Δ z L inductor in the equivalent circuit model of an infinitesimal ( Δ z λ ) TL section as shown in the margin. In addition, a shunt conductance Δ z G in parallel with capacitance Δ z C accounts in the lossy model for dielectric losses. Using perturbation theory, it can be shown that for a coax of inner and outer radii a and b , R = πσ ( 1 a + 1 b ) , while for a parallel-plate transmis- sion line of width W , R = 4 π W πσ , in terms of conductivity σ and per- meability μ of the T.L. conductors.
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