Understanding the Concept:
A quarter-wave transformer (\(\lambda/4\) transmission line section) is widely used for matching a transmission line of characteristic impedance \( Z_0 \) to a purely resistive load impedance \( Z_L \).
• The input impedance looking into a quarter-wavelength line terminated by a load \( Z_L \) is expressed as:
\[
Z_{\text{in}} = \frac{Z_t^2}{Z_L}
\]
Where \( Z_t \) is the characteristic impedance of the matching quarter-wave segment.
• For a perfect match with zero reflection, the input impedance of this matching network section must equal the characteristic impedance of the source feed line:
\[
Z_{\text{in}} = Z_0 \implies Z_0 = \frac{Z_t^2}{Z_L} \implies Z_t = \sqrt{Z_0 \cdot Z_L}
\]
Step 1: Substitute Parameters and Compute
We are given:
\[
Z_0 = 25\,\Omega
\]
\[
Z_L = 100\,\Omega
\]
Using the geometric mean relationship:
\[
Z_t = \sqrt{25 \times 100} = \sqrt{2500} = 50\,\Omega
\]
Therefore, a \( 50\,\Omega \) line provides a perfect match, matching Option (B).