S-parameters

scattering parameters. A set of parameters describing the scattering and reflection of traveling waves when a network is inserted into a transmission line. S- parameters are normally used to characterize high frequency networks, where simple models valid at lower frequencies cannot be applied. S-parameters are normally measured as a function of frequency, so when looking at the formulae for S-parameters it is important to note that frequency is implied, and that the complex gain (i.e. gain and phase) is also assumed. For this reason, S-parameters are often called complex scattering parameters.

The incident waves are designated by the letter a_n, where n is the port number of the network. For each port, the incident (applied) and reflected wave are measured. The reflected wave is designed by b_n, where n is the port number.

When the incident wave travels through the network, its value is multiplied (i.e. is gain and phase are changed) by the scattering parameter (which is analogous to the gain of the network), thus giving the resulting output value. For example, when wave a₁ travels through a network, the output value of the network is simply the value of the wave multiplied by the relevant S-parameter.

S-parameters can be considered as the gain of the network, and the subscripts denote the port numbers. The ratio of the output of port 2 to the incident wave on port 1 is designated S₂₁. Likewise, for reflected waves, the signal comes in and out of the same port, hence the S-parameter for the input reflection is designated S₁₁.

For a two port network, (assume use of matched loads)

S₁₁ is the reflection coefficient of the input
S₂₂ is the reflection coefficient of the output
S₂₁ is the forward transmission gain
S₁₂ is the reverse transmission gain (from output to input).

S-parameters can be converted to impedance by taking the ratio of (1 + S₁₁) to (1- S₁₁) and multiplying the result by the characteristic impedance (often 50 or 75 ohms).

A Smith chart is used to convert between impedance and S-parameters.

S parameters are analogous to frequency response functions, but the terms are used at high and lower frequencies, respectively.

Units:

Gain (dB or ratio) and phase (degrees or radians)

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