# Lossless quarter wavelength transmission lines

so my lecturer released a past paper with the answers and I have no idea how he got the answer to this question. Please could someone explain it to me? Thanks.

The question is: A lossless quarter-wavelength transmission line is terminated by a load of 100+j50 Ω at one end. The input impedance seen at the other end is 100-j50 Ω. What is the characteristic impedance of the quarter-wavelength transmission line?

The answer is meant to be 112Ω

• In order to perform Conjugate matching of Zin to Zout the Zo must be the RMS result of $Z_o=(x^2+y^2)^{0.5}$ for Zin=x+jy Jan 8 at 23:16
• amanogawa.com/transmission.html Jan 9 at 9:23

$$\dfrac{Z_{IN}}{Z_0}=\dfrac{Z_0}{Z_L}$$
If you rearrange that formula and solve for $$\Z_0\$$ you'll get the answer 111.803 Ω