Power divider S-parameters – Equal split and input match Assertion (A): In an ideal, lossless 3-port equal power divider, the input port is matched (S11 = 0), and the magnitudes of the transmission coefficients to the two output ports are |S21| = |S31| = 0.707 (that is, 1/√2). Reason (R): In such a power divider, all incident power at port 1 is equally divided between ports 2 and 3 while maintaining a match at the input.

Introduction:
Microwave power dividers (also called splitters) are 3-port networks that divide input power from port 1 into two outputs at ports 2 and 3. For the canonical equal-split, matched case, learners should know the key S-parameters: input match (S11 ≈ 0) and equal transmission magnitudes (|S21| = |S31| = 1/√2), corresponding to a 3 dB split when the divider is ideal and lossless. This directly connects conservation of power with symmetry in a reciprocal passive network.

Given Data / Assumptions:

• Three-port network: port 1 (input), ports 2 and 3 (outputs).
• Ideal, lossless, reciprocal, and matched at port 1 in the equal-split case.
• Equal power division between ports 2 and 3, with practical examples such as the Wilkinson divider.

Concept / Approach:

S-parameters define how incident and reflected waves relate at each port. A matched input means S11 = 0, so no power reflects back to port 1. Equal split requires |S21|^2 = |S31|^2 = 0.5, hence |S21| = |S31| = 1/√2 ≈ 0.707. These relations stem from power conservation and symmetry for an ideal equal divider; phase relations depend on topology (for instance, the Wilkinson has a specific relative phase and an isolation resistor between outputs).

Step-by-Step Solution:

Verification / Alternative check:

For a Wilkinson divider, S11 ≈ 0 (matched), |S21| = |S31| = 1/√2, and S23 ≈ 0 (isolation). Measured data across the design band confirm these magnitudes near the center frequency, validating both the assertion and its explanation.

Why Other Options Are Wrong:

Any claim denying input match or equal-split magnitudes contradicts ideal divider behavior. Saying that the reason does not explain the assertion ignores that equal split plus input match numerically yields the stated S-parameter magnitudes for a lossless reciprocal design.

Common Pitfalls:

Confusing voltage division with power division (3 dB corresponds to 0.707 in voltage magnitude, not 0.5). Another pitfall is assuming all three ports are simultaneously matched in any 3-port passive network; ideal equal-split with isolation requires a resistive network (e.g., Wilkinson) to achieve practical matching and isolation constraints.

Final Answer:

Both A and R are correct and R is the correct explanation of A.