Question 2.5: FREQUENCY RESPONSE OF A QUARTER-WAVE TRANSFORMER Consider a ...
FREQUENCY RESPONSE OF A QUARTER-WAVE TRANSFORMER Consider a load resistance \ R_{L}=100 \Omega to be matched to a \ 50 \Omega line with a quarter-wave transformer. Find the characteristic impedance of the matching section and plot the magnitude of the reflection coefficient versus normalized frequency,\ f/ f_{0}, where \ f_{0} is the frequency at which the line is \ \lambda/4 long.
From (2.63)\ Z_{1}=\sqrt{\left( Z_{0}\right)\left(R_{l}\right) } , the necessary characteristic impedance is
\ Z_{1}=\sqrt{\left(50\right)\left(100\right) } =70.71\OmegaThe reflection coefficient magnitude is given as
\ \left|\Gamma \right| =\left|\frac{Z_{in}-Z_{0}}{Z_{in}+Z_{0}} \right| ,
where the input impedance \ Z_{in}is a function of frequency as given by (2.44)
\ Z_{in}=Z_{0}\frac{Z_{L}+jZ_{0}\tan \beta l}{Z_{0}+jZ_{L}\tan \beta l} . The frequency dependence in (2.44) comes from the \ \beta _{l} term, which can be written in
terms of \ f/f_{0} as
where it is seen that\ \beta_{l}=\pi/2 for f=f_{0},as expected. For higher frequencies the matching section looks electrically longer, and for lower frequencies it looks shorter. The magnitude of the reflection coefficient is plotted versus\ f/f_{0}in
Figure 2.17.
