# Find impedance matching in a quarter wave transformer

I would like to know if there is a way you can find the impedance that has to be set between the load $Z_L$ when it is complex and the characteristic impedance $Z_0$ to have impedance matching without using the Smith Chart, since we are not allow to use it and all the methods I have search use it. It has to do something with the formula:$$Z_{in}=Z_1\frac{Z_L+jZ_1\tan\beta l}{Z_1+jZ_L\tan\beta l}=Z_0$$ and $\beta l=(2\pi/\lambda)(\lambda/4)=\pi/2$. The goal is to compute $Z_1$ and it seems always to be real. The formula can be found in Microwave Engineering, Pozar, page 79, and with a scheme like this one:

also found in page 92, but the book specifies that $R_L$ is real. The closest thing that I have found to my problem is this one: but it uses the Smith Charts as it says so. In my problem $Z_L=25-j175$ and $Z_0=36.11$, giving a result of $Z_1=\sqrt{75\cdot36.11}$ but I don't know how that result has been achieved. Could it be related to a multiple media problem?

• If you express the load in polar coordinates vs cartesian then find the conjugate length to match Zo then it is like using a Smith Chart Commented Apr 13, 2018 at 16:25
• Martin - are we done here now? Do you need any further clarification? Commented Sep 19, 2021 at 10:13

Find impedance matching in a quarter wave transformer
For a quarter wave transformer $Z_{IN} = \dfrac{Z_0^2}{Z_L}$