Assertion (A): A quarter-wave transformer is used to match a purely resistive load to a transmission line. Reason (R): A quarter-wave transformer is a transmission-line section whose electrical length is one-quarter wavelength at the design frequency. Choose the correct option about A and R.

Introduction:
Quarter-wave transformers are classic matching networks for resistive loads at a narrowband center frequency. The principle exploits impedance inversion along a line section of length lambda/4.

Given Data / Assumptions:

• Load is purely resistive at design frequency
• Transformer section length = lambda/4
• Losses are small

Concept / Approach:
The input impedance of a line of length l is Z_in = Z_t^2 / R_L when l = lambda/4 and the load is resistive, where Z_t is the transformer's characteristic impedance. Choosing Z_t = sqrt(R_L * Z0) provides a perfect match.

Step-by-Step Solution:

Verification / Alternative check:
Smith chart rotation by 180 degrees (lambda/4) inverts resistance, confirming the impedance-transform property.

Why Other Options Are Wrong:

• B: R is directly the reason quarter-wave transformers work.
• C/D: Either deny true statements or mismatch causality.

Common Pitfalls:
Applying the lambda/4 transformer to reactive or broadband loads without additional matching; ignoring frequency sensitivity.

Final Answer:
Both A and R are correct and R is the correct explanation of A