Identify device from S-matrix with S22 term

Identification

       | S11    0    S13   S14 |
 [S] = |  0    S22  -S13   S14 |
       | S13  -S13    0     0  |
       | S14   S14    0     0  |

• Ports 1 and 2 are isolated (S12 = 0); ports 3 and 4 are isolated (S34 = 0) and matched (S33 = S44 = 0).
• Port 3 gets equal and opposite-phase signals from ports 1 and 2 (S31 = −S32): port 3 is the E-arm.
• Port 4 gets equal and in-phase signals from ports 1 and 2 (S41 = S42): port 4 is the H-arm.
• The matrix is symmetric, so the device is reciprocal.

The device is a magic tee (hybrid E-H tee) with ports 1 and 2 as the collinear arms.

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

Judging the condition (lossless junction)

For a lossless device [S]*ᵀ[S] = I:

 Row 3: |S13|² + |S13|² = 1  -> |S13| = 1/√2
 Row 4: |S14|² + |S14|² = 1  -> |S14| = 1/√2
 Row 1: |S11|² + |S13|² + |S14|² = 1
        |S11|² + 1/2 + 1/2 = 1  -> S11 = 0
 Row 2: similarly            -> S22 = 0

So the given form is consistent only when S11 = S22 = 0: matching the E and H arms automatically matches the collinear arms. The device is then the ideal magic tee:

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

If S11, S22 ≠ 0 in practice, the junction is lossy or imperfectly matched.

Characteristics

• H-arm input (4): equal in-phase split to ports 1 and 2; no output at E-arm.
• E-arm input (3): equal split to 1 and 2 with 180° phase difference; no output at H-arm.
• Input at port 1: half to each of E and H arms; port 2 isolated.
• Sum/difference action: in-phase inputs at 1 and 2 add at the H-arm; anti-phase inputs add at the E-arm.
• Matched, lossless, reciprocal. Used in balanced mixers, duplexers, impedance bridges and monopulse comparators.