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Magic tee diagram and S-parameter derivation

Short answer

A magic tee combines E-plane and H-plane tees with ports 1 and 2 collinear, port 3 H-arm, and port 4 E-arm. Using symmetry S13=S23, S14=-S24, isolation S34=0, matching S33=S44=0, reciprocity, and unitarity gives S=(1/√2)[[0,0,1,1],[0,0,1,-1],[1,1,0,0],[1,-1,0,0]].

Draw a neat diagram of a magic tee and derive its S-parameter.

A magic tee is an E-plane tee and an H-plane tee joined at the same point. Ports 1 and 2 are collinear arms, port 3 is the H-arm and port 4 the E-arm.

              E-arm (4)
                | |
     -----------+ +-----------
 (1) ===========+   +=========== (2)
     -----------+ +-----------
               / /
              / /  H-arm (3)

Derivation of the S-matrix

  1. H-arm property: input at port 3 splits equally and in phase to 1 and 2: S13 = S23.
  2. E-arm property: input at port 4 splits equally and in anti-phase: S14 = −S24.
  3. Isolation of E and H arms (by geometry/symmetry): S34 = S43 = 0.
  4. Matching: E and H arms matched with posts: S33 = S44 = 0.
  5. Reciprocity: Sᵢⱼ = Sⱼᵢ.

So far:

       | S11  S12  S13  S14 |
 [S] = | S12  S22  S13 -S14 |
       | S13  S13   0    0  |
       | S14 -S14   0    0  |
  1. Lossless (unitary) condition:
 Row 3: 2|S13|² = 1           ->  S13 = 1/√2 (real, by choice of reference plane)
 Row 4: 2|S14|² = 1           ->  S14 = 1/√2
 Row 1: |S11|² + |S12|² + 1 = 1 -> S11 = S12 = 0
 Row 2: |S12|² + |S22|² + 1 = 1 -> S22 = 0

Result:

              | 0   0   1   1 |
 [S] = (1/√2) | 0   0   1  -1 |
              | 1   1   0   0 |
              | 1  -1   0   0 |

All ports are matched, ports 1–2 and 3–4 are isolated, and power fed into any port divides equally between the two ports that are not isolated from it.

✓ Answer checked by Claude Opus 5.5 (high effort)

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