Magic-tee (hybrid tee) – Identify correct S-parameter relationships For the 4-port waveguide magic tee with ports 1 and 2 co-linear, port 3 the H-arm (sum port), and port 4 the E-arm (difference port), which of the following sets are correct?

Introduction:
The magic tee (hybrid tee) combines E-plane and H-plane tees into a single 4-port junction. Understanding its ideal S-parameter symmetries is essential for using it as a power combiner/splitter or as a bridge in microwave measurements, where isolation and phase properties matter for accurate setups.

Given Data / Assumptions:

• Port 1 and 2 are the co-linear ports; port 3 is the H-arm (sum); port 4 is the E-arm (difference).
• All ports are matched ideally over a narrow band near the design frequency.
• Lossless, reciprocal network in its idealized form with symmetric geometry.

Concept / Approach:

Key properties are: (i) co-linear ports are isolated (S12 = 0), (ii) the side arms are isolated from each other (S34 = 0), (iii) coupling from a co-linear port to the H-arm is equal for ports 1 and 2 (S13 = S23), and (iv) coupling to the E-arm has a 180° phase sign difference (S14 = −S24). The provided grouped statements include these relationships; the set that is always correct and unambiguous is the one that asserts S13 = S23, S12 = 0, and S34 = 0 simultaneously.

Step-by-Step Solution:

Verification / Alternative check:

Standard S-matrices for magic tees show S12 = S34 = 0 and equal-magnitude relations noted above. Measurement benches use the E-arm for differential signals precisely because of this sign inversion.

Why Other Options Are Wrong:

Any option lacking the isolation statements or asserting S14 = S24 (missing the minus sign) is not fully correct for an ideal magic tee. Also, asserting S13 ≠ S23 contradicts symmetry.

Common Pitfalls:

Ignoring the sign inversion at the E-arm; assuming all four ports can be perfectly matched and isolated over a very wide band (in practice, these are narrowband approximations).

Final Answer:

S13 = S23; S12 = 0; S34 = 0.