A circulator is a 3-port device, transporting incoming waves lossless from port 1 to port 2, from port 2 to port 3 and from port 3 to port 1. In all other directions, there is no energy flow. The ideal circulator cannot be characterized with Z or Y parameters, because their values are partly infinite. But implementing with S parameters is practical (see equation 9.2).
With the reference impedances , and for the ports 1, 2 and 3 the scattering matrix of an ideal circulator writes as follows.
An ideal circulator is noise free.