Topic: Electromagnetics / Transmission Lines
A one wavelength long, 50 ohm line is terminated in $Z_L=30-j40$ ohms. Its input terminals are connected to the load end of a 100 ohm, quarter-wavelength line. What is the impedance seen at the input to the 100 ohm line?
- A. $30+j40$
- B. $60+j40$
- C. $120+j160$
- D. None of the above
Answer: C. $120+j160$
Explanation: A transmission line of length $n\lambda$ (where $n$ is an integer) repeats the load impedance at its input. Thus, for the $1\lambda$ long $50~\Omega$ line, its input impedance is $Z_{in1} = Z_L = 30-j40~\Omega$.
This $Z_{in1}$ acts as the load for the subsequent $\lambda/4$ line ($Z_{02} = 100~\Omega$).
For a quarter-wavelength line, the input impedance is $Z_{in} = \frac{Z_0^2}{Z_{load}} = \frac{100^2}{30-j40}$.
$Z_{in} = \frac{10000}{30-j40} = \frac{10000(30+j40)}{(30-j40)(30+j40)} = \frac{10000(30+j40)}{30^2+40^2} = \frac{10000(30+j40)}{2500} = 4(30+j40) = 120+j160~\Omega$.