Question

A lossless transmission line with characteristic impedance \(Z_0 = 50 \, \Omega\) is terminated with an unknown load. The magnitude of the reflection coefficient is \(|\Gamma| = 0.6\). As one moves towards the generator from the load, the maximum value of the input impedance magnitude looking towards the load (in \(\Omega\)) is \(\_\_\_\_\).

Hint:

To compute the maximum input impedance for a transmission line, apply the formula \(Z_{{max}} = Z_0 \frac{1 + |\Gamma|}{1 - |\Gamma|}\). This formula is particularly useful in characterizing transmission line mismatches.

Answer 1

Step 1: Calculate the input impedance.
The formula for the maximum input impedance is: \[ Z_{{max}} = Z_0 \frac{1 + |\Gamma|}{1 - |\Gamma|}, \] where \(Z_0\) is the characteristic impedance and \(|\Gamma|\) is the magnitude of the reflection coefficient. Substituting the given values, \(Z_0 = 50 \, \Omega\) and \(|\Gamma| = 0.6\): \[ Z_{{max}} = 50 \cdot \frac{1 + 0.6}{1 - 0.6}. \] Simplify: \[ Z_{{max}} = 50 \cdot \frac{1.6}{0.4} = 50 \cdot 4 = 200 \, \Omega. \] Final Answer: \[ \boxed{200 \, \Omega} \]