A transmission line is attached to an oscillating circuit and impedance matched.For technical applications it is extremely important to be able to transfer as much energy from a transmission line to some load. In this problem we will learn how this is achieved and understand the basic principles signal reflection and impedance matching.

Problem Statement

A transmission line is attached to an oscillating circuit and impedance matched.

Find out how to impedance-match a serial oscillator circuit to a transmission line! Proceed as follows:

  1. In The Transmission Line - Deriving the Telegrapher Equation we found that a transmission line is described by the telegrapher equation\[\begin{eqnarray*} \partial_{xx}U\left(x,\omega\right)&=&\left(r_{L}-\mathrm{i}\omega l\right)\left(g_{C}-\mathrm{i}\omega c\right)U\left(x,\omega\right)\\&\equiv&-k^{2}\left(\omega\right)U\left(x,\omega\right)\end{eqnarray*}\]which also holds for the current. Here, the parameters are quantities per length. In frequency space, the telegrapher equation is a one dimensional Helmholtz equation which permits propagating solutions of the form\[\begin{eqnarray*} U\left(x,\omega\right)&=&U_{0}^{+}\left(\omega\right)e^{\mathrm{i}k\left(\omega\right)x}+ U_{0}^{-}\left(\omega\right)e^{-\mathrm{i}k\left(\omega\right)x} \end{eqnarray*}\]where now the \(U_{0}^{\pm}\) are modal amplitudes. The characteristic impedance is the ratio between voltage and current for outwards propagating waves, \(Z_{0}\left(\omega\right)=U_{0}^{+}\left(\omega\right)/I_{0}^{+}\left(\omega\right)\). Using the relations between voltage and current, calculate the characteristic impedance for the transmission line.
  2. Find the general voltage reflection coefficient \(\Gamma\left(\omega\right)=U_{0}^{-}\left(\omega\right)/U_{0}^{+}\left(\omega\right)\) if the transmission line is terminated at \(x=0\) and attached to some load with impedance \(Z_{L}\left(\omega\right)=U_{L}\left(\omega\right)/I_{L}\left(\omega\right)\). Show under which condition a serial \(RLC\)-circuit attached to a lossless transmission line with \(g_{C}=r_{L}=0\) is impedance matched, i.e. can absorb all of the incoming energy.

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