Three-port S-parameters, return and insertion loss

Use of S-parameters for three-port analysis

A three-port network (tee junction, power divider, circulator) is described by a 3 × 3 S-matrix that relates outgoing waves b to incoming waves a, with each port terminated in a matched load Z0:

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

• Sᵢᵢ is the reflection coefficient at port i; Sᵢⱼ is the transmission from port j to port i.
• |Sᵢⱼ|² gives the fraction of power going from port j to port i, so power division, isolation and matching are read directly.
• Reciprocity (Sᵢⱼ = Sⱼᵢ), losslessness ([S]*ᵀ[S] = I) and matching (Sᵢᵢ = 0) can be checked from the matrix.
• Using these conditions one proves that a 3-port cannot be lossless, reciprocal and matched at all ports at once. Hence a lossless matched 3-port must be a non-reciprocal circulator, and a reciprocal lossless tee junction must have at least one mismatched port.

Example (ideal circulator): S21 = S32 = S13 = 1 and all other terms zero; power goes 1 → 2 → 3 → 1.

Return loss and insertion loss

• Return loss (RL): the ratio, in dB, of incident power to reflected power at a port. It shows how well the port is matched.

 RL = 10 log(Pin/Pref) = -20 log|S11|  dB

A large RL means a good match (e.g. |S11| = 0.1 gives RL = 20 dB); a short or open gives RL = 0 dB.

• Insertion loss (IL): the ratio, in dB, of power delivered to the load without the device to power delivered with the device inserted; for a matched network it is the loss from input to output port.

 IL = 10 log(Pin/Pout) = -20 log|S21|  dB

For example, |S21| = 0.9 gives IL = 0.92 dB.