After the mid-1900s, high frequency networks became increasingly prevalent. When analyzing low frequency circuits parameters such as voltages and currents are easily realized. From these signals, Y and Z (admittance and impedance) parameters can be used to describe a network. For the Radio Frequency and Microwave range, S parameters are much more applicable when studying a network of a single port or multiple ports. Each S parameter can be placed in an NxN square matrix where N is the number of ports. For a single port network, only the parameter S11 (also known as ᴦ (gamma or voltage reflection coefficient)) can be realized. The S parameters are unitless because they are ratios of voltages. The parameters can be viewed as both reflection and transmission coefficients for multi-port networks. S parameters with subscripts of the same number are reflection coefficients, as they describe the ratio of voltage waves at a single port (reflected to incident).
For a two-port network, only parameters S11, S12, S21, S22 exist. For a simple network like this, S11 represents return loss or reflection at port 1. S22 is the output reflection coefficient. S12 and S21 are transmission coefficients where the first subscript is the responding port and the second the incident port. For example, S21 would be the “forward gain” at port 2 incident from port 1. The following diagram shows an abstracted view of a two-port network, where each “a” and “b” are normalized by the system’s characteristic impedance. Each S parameter can be calculated by terminating a port with a matched load equal to the characteristic impedance. For example, when calculated return loss for a two-port network, port 2 should be terminated by a matched load reducing a2 to zero. For calculating S1,2 or S2,2 port 1 would be terminated with a matching load to reduce a1 to zero. Each “a” is an incident wave and “b” a reflected wave. Having a matched load at a port results none of the incident wave being reflected due to impedance mismatching.
This leads to the following voltage ratios:
An amplitude with a negative superscript indicates a reflected wave, and an amplitude with a positive superscript indicates a forward propagating wave.