Question 6.9: A lossless transmission line is terminated with an impedance...
A lossless transmission line is terminated with an impedance whose value that is ½ the characteristic impedance on the line. What impedance should you put in parallel with the line λ/4 in front of the load to minimize the reflection of the wave from the signal generator?
In order to reduce the reflection, the parallel combination of ZQ and the input impedance at this location should be equal to the characteristic impedance of the transmission line. Therefore, we write
\begin{gathered}Z_{c}=\frac{Z_{Q} Z_{\text {in }(\lambda / 4)}}{Z_{Q}+Z_{\text {in (λ/4) }}}=\frac{Z_{Q}\left(Z_{c}^{2} /Z_{L}\right)}{Z_{Q}+\left(Z_{c}^{2} / Z_{L}\right)}=\frac{Z_{Q}\left(Z_{c}^{2} /\left(Z_{c} /2\right)\right)}{Z_{Q}+\left(Z_{c}^{2} /\left(Z_{c} / 2\right)\right)}\\\\ 1=\frac{Z_{Q} 2}{Z_{Q}+2 Z_{c}} \Rightarrow Z_{Q}=2 Z_{c}\end{gathered}