Magic-tee (hybrid-T) — equivalent circuit coefficients under matched conditions If all four ports of a magic-tee are perfectly matched and isolated as per its ideal equivalent circuit, what are the magnitude coefficients at the E- and H-plane arms?

Introduction / Context:A magic-tee (hybrid-T) combines an E-plane tee and an H-plane tee, giving one sum port, one difference port, and two collinear ports. In its ideal, matched form, it behaves like a 3 dB power divider/combiner with specific voltage (or field) coupling coefficients. This question tests recognition of those ideal coefficients.

Given Data / Assumptions:

• All ports are perfectly matched and isolated as per the ideal model.
• Power is split equally between appropriate arms in the ideal case.
• Lossless operation.

Concept / Approach:In the ideal magic-tee, equal split implies each coupled arm receives half the power. Half-power corresponds to a field (or voltage) magnitude factor of 1/√2 relative to the incident wave. Numerically, 1/√2 ≈ 0.707, which serves as the normalized coefficient in the equivalent circuit or scattering matrix description.

Step-by-Step Solution:

Verification / Alternative check:From the canonical S-matrix of an ideal 3 dB hybrid, the relevant scattering magnitudes are 1/√2 with appropriate phase relationships and port isolation.

Why Other Options Are Wrong:

• 2: Would imply quadruple power, which violates conservation in a passive divider.
• Mixed 2 and 0.707: Inconsistent with symmetry and equal split.

Common Pitfalls:Confusing power ratios with voltage ratios; forgetting that 3 dB corresponds to a voltage factor of 0.707, not 0.5.

Final Answer:nE = nH = 0.707