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Hybrid tee with terminated arms

Short answer

With E-arm matched, H-arm input divides equally in phase to collinear ports; with H-arm terminated, E-arm input divides equally 180 degrees out of phase. With collinear arms terminated, E and H arms are matched and isolated, enabling balanced mixers and bridge measurements.

Using proper S-Matrices, mention the properties of a Hybrid-Tee in the situation of (a) terminated E-arm, (b) terminated H-arm and (c) terminated co-planar arms.

For an ideal hybrid (magic) tee with ports 1 and 2 collinear, port 3 the H-arm and port 4 the E-arm:

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

When a port is terminated in a matched load, no wave returns into it (aₖ = 0), so its column is dropped; its row tells how much power the load absorbs. The remaining matrix describes the reduced network.

(a) E-arm (port 4) terminated in a matched load

Remaining ports 1, 2, 3:

              | 0   0   1 |
 [S] = (1/√2) | 0   0   1 |
              | 1   1   0 |
  • Input at H-arm (3) divides equally and in phase to ports 1 and 2: the device works as an H-plane (in-phase) power divider.
  • Input at port 1: half the power goes to port 3, half (b4 = a1/√2) is absorbed in the E-arm load; port 2 receives nothing.
  • In-phase equal inputs at 1 and 2 combine fully at port 3 (power combiner); any imbalance goes to the load.

(b) H-arm (port 3) terminated in a matched load

Remaining ports 1, 2, 4:

              | 0    0    1 |
 [S] = (1/√2) | 0    0   -1 |
              | 1   -1    0 |
  • Input at E-arm (4) divides equally to ports 1 and 2 but 180° out of phase: it works as an E-plane (anti-phase) divider.
  • Input at port 1: half goes to port 4, half is absorbed in the H-arm load; ports 1 and 2 remain isolated.
  • Anti-phase equal inputs at 1 and 2 combine fully at port 4.

(c) Co-planar (collinear) arms 1 and 2 terminated in matched loads

Remaining ports 3 and 4:

       | S33  S34 |   | 0  0 |
 [S] = |          | = |      |
       | S43  S44 |   | 0  0 |
  • Both E and H arms are matched (no reflection) and completely isolated from each other (S34 = 0).
  • Power fed into either the E-arm or the H-arm is split equally into the two loads and fully absorbed.
  • This isolation between E and H arms is the basis of the balanced mixer (LO at one arm, signal at the other) and of bridge measurements: with identical loads on 1 and 2, no signal passes from H-arm to E-arm; any mismatch difference between the two loads shows up at the E-arm.

✓ Answer checked by Claude Opus 5.5 (high effort)

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