The S-parameter matrix (also called S matrix, or scattering parameters) represents the linear characteristics of RF electronic circuits and components (Figure 1). It is measured with vector network analyzers, and it describes the change of the reflected wave and the transmitted wave with respect to the incident wave of the device under test (DUT) by amplitude and phase at different frequencies.

From the S-parameter matrix, you can calculate properties such as gain, loss, phase delay, voltage standing wave ratio (VSWR), and other characteristics of linear networks. S-parameters are related to the well-known impedance (Z) and admittance (Y) matrices. S-parameters have the advantage of being easier to measure at RF frequencies because the measurement process requires finite characteristic impedances as port termination.

The S-parameter matrix can be used to describe networks with an arbitrary number of ports. In the following description, for simplicity, we consider a network with two ports as in Figure 1. a(t) and b(t) represent incident and reflected waves, respectively, at port 1 (usually called *input port*) and port 2 (usually called *output port*). The incident and reflected waves are directly related to the voltages and currents at the port terminals, as indicated by the definitions below. The characteristic impedance Z_{0} (normally 50 Ω) indicates the circuit termination used for the measurement.

The elements of the S-parameter matrix are complex and can be visualized on a Cartesian plot in terms of magnitude and phase (Figures 2 and 3), or on a polar plot. The Smith chart (Figure 4) is a special polar plot format popular for S-parameter plotting and often used for the design of input/output matching networks.

## Benefits of Using S-Parameters

S-parameters are suitable for describing high-frequency circuits and components characteristics for three reasons:

**Easier to measure than Y or Z matrices:**Other equivalent matrix descriptions, such as Y or Z, require direct measurement of currents and voltages, and open- and short-circuit terminations at the device ports. At RF and microwave frequencies, these terminating conditions cause complete reflection of the incident wave when it reaches the open- or short-circuit termination, which can cause device instability. Moreover, it is hard to guarantee the open- and short-circuit condition over a large frequency range. Because S-parameters are measured using incident and reflected waves by terminating the device ports with a finite characteristic impedance, they are not affected by any of these issues.**Easy to convert to other parameters:**Since S-parameters are directly related to Z-parameters (voltage to input/output current), Y-parameters (current to input/output voltage), and other linear matrices (T, ABCD, H) via linear transformations, they can be easily converted and subsequently used in these other formats for circuit analysis or simulation.

RF Toolbox™ provides the necessary functions to easily transform N-port S-parameters into an equivalent representation.**Flexible for analysis and simulation:**S-parameters are often stored in a standard file format called Touchstone. Most RF analysis tools and simulators can read and write Touchstone files, thus making it a portable file format to exchange measurements and design information.

Static frequency domain analysis of S-parameters is often used for the design of matching networks, and can be combined with an optimization routine to find tradeoffs between different requirements (Figure 5).

S-parameters can also be used for simulation of linear networks combined with digital signal processing algorithms to account for frequency-dependent effects, such as those in communication links.

## Basic S-Parameter Concepts

As described above, the S-parameter matrix provides a relationship between the reflected wave and the transmitted wave with respect to the incident wave of the DUT at each port and for each operating frequency.

For example, for a two-port device, you have four S-parameters representing the bidirectional behavior of the network as a function of frequency (Figure 6):

- S11 = input port reflection
- S12 = reverse gain
- S21 = forward gain (linear gain/insertion loss)
- S22 = output port reflection

From the definition, it is easy to see that S11, for example, is measured by applying an incident wave a1 to port1, and measuring the reflected wave b1 at the same port, while port2 is terminated by a load impedance identical in value to the characteristic impedance of the network. S11 is defined as the ratio of the reflected wave to the incident wave, and provides a direct measure of the matching condition of the input port (Figure 7). For example, when S11 is equal to 1, this represents an open circuit; when S11 is equal to -1, this represents a short circuit; and S11 = 0 represents a perfectly matched circuit.

## Using S-Parameters in MATLAB and Simulink

RF Toolbox and RF Blockset™ offer a wealth of functions and objects that enable you to design, model, analyze, and visualize networks using filters, transmission lines, amplifiers, mixers, and other RF components. You can easily read and write N-port Touchstone files—the standard format of S-parameters. This makes it easy to analyze RF measurement data and optimize the design of matching networks using lumped and distributed networks.

For example, RF Toolbox provides the typical functions for the conversion between S-parameters and Z, Y, ABCD, H, G, and T network parameters. Functions are also available for selecting S-parameter ports and for single-ended conversion to common mode and differential mode, which are often used for signal integrity analysis of backplanes.

RF Toolbox also provides functions for S-parameter de-embedding, cascading, and visualization to support RF test engineers in their typical tasks. By combining S-parameter data analysis tasks with the measurement process, the workflow can be automated and easily scaled up to test a wider range of operational scenarios.

Using the RF Toolbox function `rationalfit`

, you can fit S-parameters and general frequency-domain data with an equivalent Laplace transfer function, which can then be used for circuit analysis and time-domain simulation. This is particularly convenient for extracting equivalent circuit representations of RF components, the analysis of signal integrity problems, and the design of matching networks and backplane equalizers.

Data analysis and visualization of S-parameters can be easily automated and scaled up to extract statistical information on large amount of data (Figure 8).

With the RF Budget Analyzer app, you can analyze the RF budget of a transmitter or a receiver in terms of gain, power, noise figure, and third-order nonlinearity (Figure 9).

RF system designers usually start with the specifications, such as the gain of the entire system, noise figure (NF), and nonlinearity (IP3), and partition these specifications over the different stages of the RF cascade. Often, they use complicated spreadsheets to perform simple link analysis over different operating conditions. The RF Budget Analyzer app provides system engineers with a framework for budget analysis, including S-parameter mismatches, comprehensive visualization over the different stages, and a MATLAB^{®} interface for programmatically analyzing different scenarios.

The RF Budget Analyzer app enables you to directly import two-port Touchstone files to describe linear RF components. The budget analysis is performed over the signal bandwidth, accounting for input and output mismatches and thermal noise. If the Touchstone file includes measured spot-noise data, this data will be used in the budget analysis. Alternatively, if the S-parameter data is passive, the thermal noise associated with the device attenuation will be included in the analysis.

In addition, the RF Budget Analyzer app also provides the ability to automatically generate a Simulink^{®} model of the chain using the Circuit Envelope simulation technology of RF Blockset™ (Figure 10). The generated model and verification test benches can be used for the simulation and validation of the chain behavior using modulated waveforms, adding interfering signals, and modeling other imperfections that cannot be easily be analytically estimated.

In MATLAB and Simulink, you can model entire wireless communications systems by connecting RF transmitters and receivers to baseband processing algorithms (Figure 11); for example, using standard-compliant modulated waveforms such as LTE or WiFi. You can estimate the system performance in terms of bit error rate (BER) or error vector magnitude (EVM). The S-parameter data is simulated in the time domain by the Circuit Envelope solver, using either the rational fitting or the convolution-based approach.