Matched multiport networks – S-matrix condition for all ports matched A passive microwave junction is said to be matched at all of its ports. What property must its scattering (S) matrix have?

Introduction:
S-parameters describe how incident and reflected waves relate at each port of an RF network. A port is matched when a wave incident on that port produces no reflection. This condition is compactly expressed in the S-matrix as a requirement on its diagonal entries.

Given Data / Assumptions:

• Linear, time-invariant microwave network with N ports.
• Reference impedances chosen equal to the intended match (typically 50 Ω).
• Definition: matched port → S_ii = 0.

Concept / Approach:

The i-th diagonal entry S_ii equals the reflection coefficient seen looking into port i when all other ports are terminated in their reference impedances. “Matched at all ports” means S₁₁ = S₂₂ = … = S_NN = 0. Off-diagonal entries determine transmission/isolation properties, but they do not affect the matched condition itself.

Step-by-Step Solution:

Verification / Alternative check:

For a Wilkinson power divider designed for 50 Ω at all ports, the measured |S₁₁|, |S₂₂|, |S₃₃| are near zero across the design band, illustrating the condition.

Why Other Options Are Wrong:

Equal non-zero diagonals still imply reflections; “purely complex” is irrelevant; Hermitian pertains to reciprocity and losslessness only under special cases; unitarity (lossless) does not imply zero diagonals.

Common Pitfalls:

Confusing “lossless” (unitary S) with “matched.” A device can be lossless yet unmatched, or matched yet lossy.

Final Answer:

all diagonal elements are zero.