S-matrices of E-plane tee, H-plane tee and duplexer

For each junction, port numbering is: ports 1 and 2 are the collinear arms and port 3 is the side arm. A "perfectly working" device means a lossless junction with its side arm matched (for tees) or all ports matched (for the duplexer).

1. E-plane tee (series tee)

The side arm is in the plane of the E-field (on the broad wall). A wave fed into port 3 splits equally into ports 1 and 2 with 180° phase difference.

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

Check: each row has |S|² sum = 1/4 + 1/4 + 1/2 = 1 and rows are orthogonal, so it is lossless; S33 = 0 (matched side arm); S13 = −S23 (anti-phase split). Uses: power division with phase reversal, balanced mixers, part of the magic tee, impedance tuners (with a shorted side arm).

2. H-plane tee (shunt tee)

The side arm is in the plane of the H-field (on the narrow wall). A wave fed into port 3 splits equally into ports 1 and 2 in phase.

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

Again lossless and reciprocal, S33 = 0, and S13 = S23 (in-phase split). Uses: in-phase power dividers and combiners, feeding antenna arrays, part of the magic tee.

3. Duplexer (3-port circulator type)

A duplexer lets one antenna be shared by a transmitter (port 1), antenna (port 2) and receiver (port 3): transmitter power goes only to the antenna, and the received echo goes only to the receiver. A perfect duplexer is an ideal circulator:

       | 0  0  1 |
 [S] = | 1  0  0 |       TX(1) -> ANT(2) -> RX(3)
       | 0  1  0 |

• Matched at all ports, lossless (unitary), and non-reciprocal (S21 = 1 but S12 = 0), made with a magnetized ferrite.
• Transmitter → antenna: b2 = a1. Antenna → receiver: b3 = a2. Transmitter → receiver: S31 = 0 (isolation protects the receiver).

   TX (1) ---->+-----+----> ANT (2)
               | (→) |
               +--+--+
                  |
                  v
               RX (3)

Uses: radar and communication transceivers using a single antenna, isolators (with port 3 terminated), reflection amplifiers.

Note: the two tees are reciprocal and cannot have all three ports matched; only the non-reciprocal circulator can be lossless and matched at all ports.