S-parameters in Circuits and Antennas

Last edited: 2024-10-30 15:27:23

S-parameters, short for scattering parameters, are a useful and easy way to measure a network's response to input at its terminals (ports). With the S-parameters of a network determined, the behavior of the network in a new external environment can be determined. What is measured is the reflected and transmitted waves going through the network's ports.

S-parameter Formula

To calculate the S-parameters of a network one can use the normalized incident complex voltage wave $a_i$ at the $i$:th port defined as

$$a_i = \frac{V_i + Z_i I_i}{2 \sqrt{|\text{Re} Z_i|}}$$

and the normalized reflected complex voltage wave $b_i$ defined as

$$b_i = \frac{V_i - \overline{Z_i} I_i}{2 \sqrt{|\text{Re} Z_i|}},$$

where $V_i$ is the terminal voltage, $I_i$ the terminal current and $Z_i$ is an arbitrary reference impedance (and $\overline{Z_i}$ is it's complex conjugate). Then the S-parameter $S_{ij}$ is defined as

$$S_{ij} = \frac{b_i}{a_j}.$$

S-parameters For a 2-port Network

For a 2-port network, seen in the figure above, there are four different S-parameters. These can be used to describe the relation between incident waves $a_1$ and $a_2$ and the reflected waves $b_1$ and $b_2$ in the following

$$\left\{ \begin{array}{cc} b_1 = & S_{11} a_1 + S_{12} a_2 \\ b_2 = & S_{21} a_1 + S_{22} a_2 \end{array} \right. .$$

To get $S_{11}$ the second port has to be terminated with a matched load $Z_L = Z_0$ and $a_2 = 0$ so

$$S_{11} = \left. \frac{b_1}{a_1} \right|_{a_2 = 0}.$$

To get $S_{22}$ the first port has to be terminated with a matched load $Z_S = Z_0$ and $V_S = 0$ so

$$S_{22} = \left. \frac{b_2}{a_2} \right|_{a_1 = 0}.$$

To get $S_{21}$ the second port has to be terminated with a matched load $Z_L = Z_0$ and $a_2 = 0$ so

$$S_{21} = \left. \frac{b_2}{a_1} \right|_{a_2 = 0}.$$

To get $S_{12}$ the first port has to be terminated with a matched load $Z_S = Z_0$ and $a_1 = 0$ so

$$S_{12} = \left. \frac{b_1}{a_2} \right|_{a_1 = 0}.$$

Using these formulas and by choosing a real reference impedance $Z_0$ (so $Z_0=\overline{Z_0}$ we get

$$S_{11} = \frac{b_1}{a_1} = \frac{\frac{V_1}{I_1} - Z_0}{\frac{V_1}{I_1} + Z_0} = \frac{Z_1 - Z_0}{Z_1 + Z_0}.$$

Therefore we also get

$$Z_1 = Z_0 \frac{1 + S_{11}}{1 - S_{11}},$$

where at port 1, $Z_1 = \frac{V_1}{I_1}$ is the input impedance.

S-parameters For Antennas

S-parameters for an $N$:th element antenna array is calculated by

$$S_{mn} = \left. \frac{V_m^-}{V_n^+} \right|_{V_k^+ = 0 \text{ for } k \neq n}$$

where $V_m^+$ are the incident and $V_m^-$ the reflected voltage wave amplitudes for the $m$:th array element seen in the figure above.