## Impedance Matching of Transmission Lines and Oscillator Circuits

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

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.