S-parameters and three-port network S-matrix

Why S-parameters are used at microwave frequencies

At microwave frequencies the Z, Y, h and ABCD parameters become hard to use because:

• They need ideal open or short circuits at the ports, which are hard to make at high frequency (stray capacitance and inductance, radiation from an open end).
• Active devices (transistors) often oscillate or are damaged when terminated in an open or short.
• Voltage and current are not uniquely defined in a waveguide, but incident and reflected power waves are.

S-parameters are measured with all ports terminated in a matched load (Z0), they relate incident and reflected waves directly, and they are measured easily with a vector network analyzer. Their magnitudes give power flow (|Sij|² = fraction of power), so return loss, insertion loss and isolation are read off directly.

S-matrix of a 3-port network

For a 3-port network, let aᵢ be the normalized incident wave and bᵢ the reflected (outgoing) wave at port i. Then

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

• Sᵢᵢ = bᵢ/aᵢ (all other ports matched): reflection coefficient at port i.
• Sᵢⱼ = bᵢ/aⱼ (all ports except j matched): transmission coefficient from port j to port i.

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

Properties of the S-matrix of a 3-port network

1. Reciprocity (symmetry): for a passive network without ferrites or plasma, Sᵢⱼ = Sⱼᵢ, i.e. [S] = [S]ᵀ.
2. Lossless (unitary) property: [S]ᵀ[S] = [I], so for each column Σₖ|Sₖᵢ|² = 1, and for two different columns Σₖ SₖᵢSₖⱼ = 0.
3. Matched ports: if all ports are matched, S11 = S22 = S33 = 0.
4. Phase shift property: if the reference plane at port n is moved outward (away from the junction) by a length lₙ, Sᵢⱼ is multiplied by e^(−jβ(lᵢ + lⱼ)).
5. Key theorem: a 3-port network cannot be lossless, reciprocal and matched at all ports at the same time. If it is lossless and matched, it must be non-reciprocal (a circulator); if it is reciprocal and lossless, at least one port is mismatched (e.g. a tee junction).

Example 1: H-plane tee (reciprocal, lossless, port 3 matched)

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

Power fed at port 3 divides equally and in phase between ports 1 and 2; ports 1 and 2 are not matched (S11 = 1/2).

Example 2: ideal circulator (lossless, matched, non-reciprocal)

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

All ports are matched and the matrix is unitary but not symmetric, as the theorem requires.