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Three-port model and magic tee with shorted arms

Short answer

A three-port network is characterized by S parameters: Sii is reflection, Sij transmission, reciprocity gives Sij=Sji, and losslessness gives unitarity. With both E and H arms shorted at their reference planes, the reduced two-port S matrix is [[-1,0],[0,-1]], so each collinear port reflects fully with 180 degree phase shift; moving the shorts can make it a switch or phase shifter.

Design a model of a three-port network and define its characteristic parameters. Prepare the S-Matrix of a rectangular magic tee having shorted both E and H-arms and explain how it behaves.

Three-port network model and its parameters

A three-port network (e.g. an E-plane or H-plane tee, circulator or power divider) is described by its scattering matrix, relating outgoing waves b to incident waves a:

            port 3
              |
          +---+---+
 port 1 --|  [S]  |-- port 2
          +-------+

 | b1 |   | S11  S12  S13 | | a1 |
 | b2 | = | S21  S22  S23 | | a2 |
 | b3 |   | S31  S32  S33 | | a3 |

Characteristic parameters:

  • Sᵢᵢ: reflection coefficient at port i (return loss = −20 log|Sᵢᵢ|).
  • Sᵢⱼ: transmission from port j to i (insertion loss = −20 log|Sᵢⱼ|).
  • Reciprocal: Sᵢⱼ = Sⱼᵢ. Lossless: [S]*ᵀ[S] = I. Matched: Sᵢᵢ = 0.
  • A 3-port cannot be lossless, reciprocal and fully matched at the same time.

Example, H-plane tee: S = [[1/2, −1/2, 1/√2], [−1/2, 1/2, 1/√2], [1/√2, 1/√2, 0]].

Magic tee with both E and H arms shorted

Ideal magic tee (1, 2 collinear; 3 = H-arm; 4 = E-arm):

 b1 = (a3 + a4)/√2      b3 = (a1 + a2)/√2
 b2 = (a3 - a4)/√2      b4 = (a1 - a2)/√2

A short at the reference plane of an arm reflects the wave with Γ = −1, so a3 = −b3 and a4 = −b4:

 a3 = -(a1 + a2)/√2,   a4 = -(a1 - a2)/√2

 b1 = (a3 + a4)/√2 = -(2a1)/2 = -a1
 b2 = (a3 - a4)/√2 = -(2a2)/2 = -a2

So the reduced two-port (ports 1 and 2) has

       | -1   0 |
 [S] = |        |
       |  0  -1 |

Behaviour: each collinear arm behaves like a short circuit: all power entering port 1 (or 2) is reflected back with 180° phase, and nothing is transmitted to the other arm. The power entering port 1 splits into the E and H arms, is reflected by the shorts, and recombines entirely at port 1.

General case (shorts at distances l3 and l4): Γ3 = −e^(−j2βl3), Γ4 = −e^(−j2βl4). Then

 b1 = ((Γ3 + Γ4)/2)·a1 + ((Γ3 - Γ4)/2)·a2
 b2 = ((Γ3 - Γ4)/2)·a1 + ((Γ3 + Γ4)/2)·a2

If the shorts differ in position by λg/4 (Γ4 = −Γ3), then S11 = 0 and |S21| = 1: all power passes from port 1 to port 2. By moving the shorts, the tee can act as a variable reflector, switch or phase shifter; this property is used in magic-tee phase shifters and duplexers.

✓ Answer checked by Claude Opus 5.5 (high effort)

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