## Scattering Parameters

RF/Photonics Lab

Jared Alves

November 2019

**Scattering Parameters**

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 S_{11 }(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 S_{11}, S_{12}, S_{21}, S_{22} exist. For a simple network like this, S_{11 }represents return loss or reflection at port 1. S_{22 }is the output reflection coefficient. S_{12} and S_{21 }are transmission coefficients where the first subscript is the responding port and the second the incident port. For example, S_{21 }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 S_{1,2 }or S_{2,2 }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.

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