Three-port directional coupler using S-parameters

A three-port directional coupler is analysed with the 3 × 3 S-matrix, using port 1 = input, 2 = through, 3 = coupled.

Step 1 – can a lossless, reciprocal, fully matched 3-port exist? Suppose S11 = S22 = S33 = 0 and Sᵢⱼ = Sⱼᵢ:

       |  0   S12  S13 |
 [S] = | S12   0   S23 |
       | S13  S23   0  |

Unitary conditions:

 |S12|² + |S13|² = 1
 |S12|² + |S23|² = 1
 |S13|² + |S23|² = 1
 S13*·S23 = 0,  S23*·S12 = 0,  S12*·S13 = 0

The last three need at least two of S12, S13, S23 to be zero, which then breaks one of the first three. So no lossless, reciprocal, matched 3-port coupler exists. (If reciprocity is dropped, a lossless matched 3-port is possible: that is the circulator.)

Step 2 – practical 3-port coupler. It is a 4-port coupler whose isolated port is terminated internally in a matched load. With the ideal 4-port coupler (S12 = α, S13 = jβ, α² + β² = 1), deleting port 4:

       |  0    α    jβ |
 [S] = |  α    0    0  |
       | jβ    0    0  |

It is matched and reciprocal but lossy (power entering port 2 or 3 is partly absorbed in the internal load). Coupling C = −20 log β dB; e.g. β = 0.316 gives a 10 dB coupler.