A twin-lead feeder used to connect an antenna to a receiver has

PHYSICS · Multi-part · 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 twin-lead feeder used to connect an antenna to a receiver has, per unit length, series inductance L′=1.0 μH m−1L' = 1.0\,\mu\mathrm{H\,m^{-1}} and shunt capacitance C′=40 pF m−1C' = 40\,\mathrm{pF\,m^{-1}}. Treat the line as lossless. ((a)) Model the line as a ladder of sections each of length dz\mathrm{d}z, with series inductance L′ dzL'\,\mathrm{d}z and shunt capacitance C′ dzC'\,\mathrm{d}z. Write the voltage law across one ladder section and the current law at its node, divide by dz\mathrm{d}z, and take the limit dz→0\mathrm{d}z \to 0 to obtain a pair of first-order partial differential equations in v(z,t)v(z,t) and i(z,t)i(z,t). Eliminate ii to reach ∂2v∂z2=L′C′∂2v∂t2\dfrac{\partial^2 v}{\partial z^2} = L'C'\dfrac{\partial^2 v}{\partial t^2}, and check by substitution that any v(z,t)=f(z−vpt)v(z,t) = f(z - v_pt) solves it, identifying vpv_p. ((b)) Calculate the characteristic impedance Z0=L′/C′Z_0 = \sqrt{L'/C'}, the propagation velocity vpv_p, and the time taken for a step to travel along 20 m20\,\mathrm{m} of this line.

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