What are the functions and principles of S-parameters (Part 1)?

As design frequencies now routinely go into the hundreds of MHz and tens of GHz range, conventional voltage and current measurements are not useful. However, scattering (s) parameters can fully characterize the RF component or path performance.

For engineers with experience focused from DC to several hundred megahertz, it is usually sufficient to characterize components by their voltage and current characteristics for much of the design effort. However, as designs now routinely go to much-higher frequencies, often into the tens of gigahertz, this characterization is impractical, inadequate, and even misleading.

Instead, a measurement and characterization metric called scattering parameters—often called S-parameters—is used. This FAQ will look at the basics and the need for S-parameters as well as their application and test. By their nature, S-parameters and what they represent is a complicated, math-intensive topic. We will only present the basics here.

Q: When I studied basic transistors, there were h-parameters that characterized those components. Aren’t those enough for RF/microwaves?

A: Hybrid parameters (also known as h-parameters) use voltage and current ratios to represent the relationship between those two metrics in a basic two-port network (Figure 1). Remember that many components and circuits can be considered “black-box” two-port networks where all we see is the applied input and the measured output, rather than the internal device specifics and physics.

Figure 1: H-parameters are widely used for analyzing the voltage and current relationships in devices such as transistors but fall short for analysis of higher-frequency operation. (Image: Electrical4U)

The top-level equations which are associated with this model are:

Q: So, why do we need other parameters?

A: Whether you call it RF or microwaves, measuring voltage or current at those higher frequencies is tough. Even if you do, the measured values have little immediate meaning since the RF world is one of “power” and “energy” more than voltage and current. The idea of S-parameters is to avoid the need to “perfectly” model an RF circuit or component with all the inherent subtleties, strays, and parasitics—which are often unknown and even unknowable—and then do a classic Spice-like analysis based on voltage, current, impedance, and frequency.

Not only would such a model and its associated analysis be complicated and time-consuming, but it would also not be accurate in simplifying the model versus reality. Instead, S-parameters view the component or circuit as a black box with unknown internal models that can be assessed and designed solely by the input-versus-output characteristics in multiple directions.

Q: So, what do S-parameters indicate?

A: S-parameters are complex numbers (numbers with real and imaginary parts) that can be used directly or in a matrix to show reflection/transmission characteristics (amplitude and optionally phase) in the frequency domain. When a complex time-varying signal is passed through a linear network, the amplitude and phase shifts can dramatically distort the time-domain waveform. Therefore both amplitude and phase information in the frequency domain are important. S-parameters are the parameter that supports both information and has many advantages for high-frequency device characterization.

Q: How are they assessed?

A: As with h-parameters, it begins with a two-port network. The output is not short- or open-circuited as it is with h-parameters (it is challenging to create a “perfect” short or open circuit at higher frequencies). Instead, it is terminated with the characteristic impedance of the component or circuit, usually but not always 50 Ω. (Note that most RF work is at 50 Ω, but for various reasons, cable TV and related technologies are usually at 75 Ω; why that is so is an interesting story for another time).

Q: It still seems like voltage and current would be more useful, aren’t they?

A: S-parameters and the resultant scattering matrix is an arrangement that quantifies how RF energy propagates through a multi-port network. It allows accurate characterization and description of the properties of components and circuits, including their many parasitic and stray values, as simpler “black boxes.” This avoids the need to try to model the impedances, resistors, capacitances, and inductances of the circuit at the higher frequencies, where such modeling would be very difficult and likely miss many real-world subtleties.

Q: What are the specifics of S-parameters?

A: There are four S-parameters for the two-port network: S11, S12, S22, and S21 (they are sometimes subscripted, sometimes not; there’s a lot of inconsistency on this). The numbering convention for S-parameters is that the first number following the “s” is the port where the signal emerges, and the second number is the port where the signal is applied (Figure 2). For example, S21 is a measure of the signal coming out of port 2 relative to the RF stimulus entering port 1. When the numbers are the same (e.g., S11), it indicates a reflection measurement where the input and output ports are the same. S parameters can also be expressed in dB, as 20 log₁₀|Sxy| where x and y = 1 or 2.

Figure 2: The principle of S-parameters is to model the component or circuit as a two-port “black box” and then measure the signal flow (magnitude and phase) from one port to another as well as back to the source port, thus creating four distinct metrics. (Image: Keysight Technologies)

Q: How does this provide insight into the attributes of the RF device under test (DUT)?

A: The amplitude and phase information quantifies the reflection and transmission characteristics of the DUT. Some commonly measured factors are magnitude only (scalar), while others are vectors (both magnitude and phase). For example, return loss is a scalar reflection measurement, while impedance results from a vector reflection measurement. Others, such as group delay, are solely phase-related measurements.

Note that the S-parameters are a function of the frequency of the device or circuit being assessed; a single set of four numbers (real or complex) does not characterize the situation. Instead, the test result is a graph (or table) for each of the four parameters as a function of frequency (Figure 3).

Figure 3: The S11 parameter value versus frequency of a specific component; here, a detection antenna array (Image: Researchgate)

Q: What are some of the performance attributes associated with S-parameters?

A: Reflection parameters S11/S22 indicate reflection (return loss, often denoted by Greek upper-case tau τ), impedance, admittance, and VSWR. Transmission parameters S21/S12 show gain/loss (insertion loss), phase, and group delay (delay time).

Q: Is that all there is to S-parameters?

A: Absolutely not. Due to their usefulness in characterizing devices and circuits in the higher-frequency spectrum, S-parameters are the subject of detailed analysis and mathematical operations, topics covered in many textbooks and references. The math can get fairly complicated with matrices and matrix operations using complex (real and imaginary) numbers. Further, S-parameters are a frequency-domain characterization, so many aspects of design and analysis require time-domain insight, so the correlation between the two domains must be established.

Q: The two-port model used to develop the s-parameter concept has one input port and one output port, representing single-ended systems. However, many RF designs now use differential, balanced components and circuitry, in effect having two input and output ports. Can S-parameters handle this?

A: There are extensions to the basic s-parameter arrangement, which expands the two-port model to a four-port model. Of course, the analysis and math get more complicated, but it works and is useful. The analysis has four stimulus/response quadrants and creates 16 S-parameters. Matrix representation and math are essential at this stage to keep everything organized and do the subsequent analysis.

Q: I’m just curious: are S-parameters relatively new? It seems like that might be the case.

A: It depends on what is meant by “new,” of course. There were some allusions to the concept in work done in the 1920s. Still, the serious s-parameter efforts began at the MIT Radiation Laboratory (an RF/wireless lab, not an atomic-research lab) around 1948. There was also work done at Bell Labs (of course), which became well-known in the mid-1960s. The other significant development was the introduction in the 1960s of network analyzers by Hewlett Packard (subsequently known as Agilent Technologies and now as Keysight Technologies), which could measure S-parameters.

Part 2 of this FAQ looks at the measurement and instrumentation of S-parameters, and their link to the time domain and Smith chart.

EE World Online References

Impedance matching and the Smith chart, Part 1
Impedance matching and the Smith Chart, Part 2
Printed Circuit Boards, Part 4: Beyond FR-4
Passive microwave components, Part 1: isolators and circulators
Passive microwave components, Part 2: couplers and splitters
Fast interconnect analyzer reveals time- and frequency-domain details in a single acquisition
Load pull for RF devices, Part 2: How and where
Low-Cost Vector Network Analyzer covers up to 6 GHz
VNAs get updated time domain, eye diagram tools