Understanding S-parameters

What are S parameters?

S parameters are a way of representing the RF network characteristics. They describe how much of a signal is reflected back, transmitted through or transferred between the ports of a network.

An S parameter is denoted with the letter “S” followed by a pair of subscripts, where the first subscript indicates the output port and the second subscript indicates the input port. For example:

• if the energy enters through port 1 and emerges from port 1, the S parameter is named S11.
• If the energy enters through port 1 and emerges from port 2, the S parameter is named S21.
• If the energy enters through port 2 and emerges from port 3, the S parameter is named S32, and so on.

A two port network has four S parameters:

• S₁₁ is the input reflection coefficient. It measures the portion of the input signal at port 1 that is reflected back to port 1. This parameter indicates how much of the input signal is reflected due to an impedance mismatch.
• S₂₁ is the forward transmission coefficient. It measures the portion of the input signal at port 1 that is transmitted to port 2. This parameter indicates the efficiency of signal transmission from the input to the output.
• S₁₂ is the reverse transmission coefficient. It measures the portion of the input signal at port 2 that is transmitted to port 1. This parameter indicates how well port 1 is isolated from signals entering port 2.
• S₂₂ is the output reflection coefficient. It measures the portion of the input signal at port 2 that is reflected back to port 2. This parameter indicates how much of the output signal is reflected due to impedance mismatch at the output.

Representing S parameters

S parameters are complex values – meaning they have both magnitude and phase components. The magnitude indicates how much of the signal is transmitted or reflected, while the phase indicates how the timing of the signal is altered. Both components are necessary for a complete description of how the signal behaves.

S parameters can be represented by N-by-N matrices, where “N” is the number of ports in the network. Each element of the matrix, Sxy, represents the scattering parameters from port y to port x. Since Sxy is a complex number, the matrix provides:

• A structural layout of the network’s inter port interactions
• Detailed amplitude and phase information for each interaction

Another way to represent S parameters is with a Smith chart, a graphical tool primarily used to represent reflection coefficients. The Smith chart helps visualize how the impedance of a network varies with frequency. Plotting the real and imaginary components of the reflection coefficients on a Smith chart provides a clear picture of impedance matching, which is crucial for minimizing reflections and optimizing power transfer.

S parameters can be cascaded to predict the overall response of a series of connected networks. In other words, when two or more networks are connected in series, their individual S parameters can be combined mathematically to determine the total S parameters of the combined network. The ability to cascade S parameters and predict system behavior relies on their complex representation and matrix format, which enable the analysis of more complex, multi stage systems.