Bert Simonovich's Design Notes

Originally published in Signal Integrity Journal Magazine, July 2020

Signal Integrity (SI) engineers almost always have to work with S-parameters. If you haven’t had to work with them yet, then chances are you will sometime in your SI career. As speed moves up in the double-digit GB/s regime, many industry standards are moving to serial link-based architectures and are using frequency domain compliance limits based on S-parameter measurements.

A vector network analyzer (VNA) is the test instrument of choice to measure S-parameters from a device under test (DUT). By definition, each S-parameter (S_ij) is the ratio of the sine wave voltage coming out of a port to the sine wave voltage that was going in to a port (Equation 1). Each S-parameter is complex with a magnitude and a phase.

Equation 1

Sufficed to say, for mathematical reasons, the indexes refer to the port in which the voltages are coming or going. This is counter intuitive to our normal train of thought and is important to be cognisant of this relationship when working with S-parameters.

Single-ended S-parameters

Figure 1 shows an example of a 1-Port, 2-Port and 4-Port DUTs and their respective S-parameter matrices representing uniform transmission lines with respective port index labelling. Each S-parameter in the matrix are single-ended measurements from one port to another.

A 1-Port DUT has one S-parameter (S₁₁) shown in red. It is the ratio of the voltage coming out of Port 1 to the voltage going into Port 1. As a measure of reflected energy out of Port 1, it is also known as return loss (RL)

A 2-Port DUT has 4 S-parameters shown in blue. S-parameters with the same index subscript numbers, i.e. S_11, S₂₂ are RL. S-parameters with alternate index subscript numbers, are a measure of transmitted energy and is the ratio of the voltage coming out of a Port to the voltage going into the opposite Port. It is also known as insertion loss (IL). For example, S₁₂ is the ratio of the voltage coming out of Port 1 to the voltage going into Port 2, whereas S₂₁ is the ratio of the voltage coming out of Port 2 to the voltage going into Port 1.

Figure 1 From left to right examples of 1-Port (Red), 2-Port (Blue), 4-Port (Black) DUTs and their respective S-parameter matrices.

A 4-Port DUT has 16 S-parameters, divided into 4 quadrants, shown in black. As you can see the number of S-parameter combinations is the square of the number of ports. In this example, the top left quadrant 1 and bottom right quadrant 4 are the same as individual 2-Port DUTs with different port indices. They are described as:

Quadrant 1:

• S₁₁ is the ratio of the voltage coming out of Port 1 to the voltage going into Port 1. It is the RL out of Port 1.
• S₁₂ is the IL and is the ratio of the voltage coming out of Port 1 to the voltage going into Port 2. It is the IL from Port 2 to Port 1.
• S₂₁ is the ratio of the voltage coming out of Port 2 to the voltage going into Port 1. It is the IL from Port 1 to Port 2. For a uniform transmission line, S₂₁ = S₁₂.
• S₂₂ is the ratio of the voltage coming out of Port 2 to the voltage going into Port 2. It is the RL out of Port 2. For a uniform transmission line, S₂₂ = S₁₁.

Quadrant 4:

• S₃₃ is the ratio of the voltage coming out of Port 3 to the voltage going into Port 3. It is the RL out of Port 3
• S₃₄ is the ratio of the voltage coming out of Port 3 to the voltage going into Port 4. It is the IL from Port 4 to Port 3
• S₄₃ is the ratio of the voltage coming out of Port 4 to the voltage going into Port 3. It is the IL from Port 3 to Port 4. For a uniform transmission line, S₄₃ = S₃₄.
• S₄₄ is the ratio of the voltage coming out of Port 4 to the voltage going into Port 4. It is the RL out of Port 4. For a uniform transmission line, S₄₄ = S₃₃

S-parameters in the top right quadrant 2 and bottom left quadrant 3 describe the near-end and far-end coupling of the respective ports. When unwanted coupling happens at the near-end, it is referred to as near-end cross talk, or NEXT. When it happens at the far-end, it is known as far-end crosstalk, or FEXT.

Quadrant 2:

• S₁₃ is the ratio of the voltage coming out of Port 1 to the voltage going into Port 3. It is the coupling or NEXT from Port 3 to Port 1.
• S₁₄ is the ratio of the voltage coming out of Port 1 to the voltage going into Port 4. It is coupling or FEXT from Port 4 to Port 1.
• S₂₃ is the ratio of the voltage coming out of Port 2 to the voltage going into Port 3. It is coupling or FEXT from Port 3 to Port 2.
• S₂₄ is the ratio of the voltage coming out of Port 2 to the voltage going into Port 4. It is coupling or NEXT from Port 4 to Port 2.

Quadrant 3:

• S₃₁ is the ratio of the voltage coming out of Port 3 to the voltage going into Port 1. It is the coupling or NEXT from Port 1 to Port 3.
• S₃₂ is the ratio of the voltage coming out of Port 3 to the voltage going into Port 2. It is coupling or FEXT from Port 2 to Port 3.
• S₄₁ is the ratio of the voltage coming out of Port 4 to the voltage going into Port 1. It is coupling or FEXT from Port 1 to Port 4.
• S₄₂ is the ratio of the voltage coming out of Port 4 to the voltage going into Port 2. It is coupling or NEXT from Port 2 to Port 4.

Although there is no industry standard for labeling a 4 or more port DUT, a practical way is to use the port order shown so that the 2-Port DUT is a subset of the top left quadrant of the 4-Port DUT. When you do this, the port order labeling is consistent as you increase the number of ports; with odd ports on the left and even ports on the right. S₁₂ and S₂₁ always describe the IL terms; while S₁₃ and S₃₁ define the NEXT terms.

But sometimes 3^rd party 4-port S-parameters are labeled with ports 1 and 2 are on the left side, while ports 3 and 4 are on the right side. In this configuration, S₃₁ and S₄₂ are now the IL terms. This is counter intuitive when moving from 2-Port to 4 or more Port DUT and leading to potential confusion when cascading S-parameters to build a channel model, or converting to mixed-mode S-parameters. Whenever you get S-parameter files from 3^rd party, it is always prudent to test it and compare IL plots against port order to ensure you are using them correctly.

Typically, 4-port S-parameters are saved in Touchstone format with a .snp extension, where n is the number of ports. Many Electronic Design Automation (EDA) and circuit simulation software tools allows you to view and plot S-parameters from Touchstone files.

Figure 2 is a schematic of a 4-port S-parameter component used in Keysight ADS. When the component is linked to appropriate .s4p touchstone file and ports connected as shown, the 16-port S-parameter matrix can be plotted and analyzed.

Figure 2 Keysight ADS schematic used to plot 4-Port single-ended S-parameters.

The 1-port and 2-port S-parameters are included in the same plot as the 4-port S-parameters plotted in Figure 3. The top left (red) and bottom right (green) quadrants plot the return loss (RL) and insertion loss (IL), while the top right (blue) and bottom left (magenta) quadrants plot the NEXT and FEXT.

Figure 3 An example of 4-Port S-parameter single-ended plots of a uniform transmission line.

Mixed-mode S-parameters

SI engineers often have to check channel models and S-parameter measurements against industry standard compliance plots. Many of those plots are in terms of mixed-mode S-parameters, which means the single-ended measurements need to be converted to mixed-mode matrix.

Two single-ended transmission lines with coupling are also known as a differential pair, as shown in Figure 4. When we talk about single-ended transmission lines with coupling, we are usually interested in their single-ended properties like characteristic impedance (Zo), phase delay, and NEXT/FEXT relationships as described above.

But when we talk about a differential pair, we are interested in the mixed-mode S-parameters like differential and common signals and how they interact within the pair. Because we are describing the exact same interconnect, they are equivalent.

When describing a differential pair, there are only four possible outcomes in response to an input signal as defined by the mixed-mode S-parameter matrix:

• A differential signal enters the differential pair and a differential signal comes out
• A differential signal enters the differential pair and a common signal comes out
• A common signal enters the differential pair and a differential signal comes out
• A common signal enters the differential pair and a common signal comes out

Figure 4 Single-ended vs mixed-mode S-parameter matrices of two coupled transmission lines.

Mixed-mode S-parameters in each quadrant are described as:

SDD Quadrant (Red):

• SDD₁₁ is the ratio of the differential signal coming out of Port 1 to the differential signal going into Port 1. It is the differential RL out of Port 1.
• SDD₁₂ is the ratio of the differential signal coming out of Port 1 to the differential signal going into Port 2. It is the differential IL from Port 2 to Port 1.
• SDD₂₁ is the ratio of the differential signal coming out of Port 2 to the differential signal going into Port 1. It is the differential IL from Port 1 to Port 2.
• SDD₂₂ is the ratio of the differential signal coming out of Port 2 to the differential signal going into Port 2. It is the differential RL out of Port 2.

SDC Quadrant (Blue):

• SDC₁₁ is the ratio of the differential signal coming out of Port 1 to the common signal going into Port 1.
• SDC₁₂ is the ratio of the differential signal coming out of Port 1 to the common signal going into Port 2.
• SDC₂₁ is the ratio of the differential signal coming out of Port 2 to the common signal going into Port 1.
• SDC₂₂ is the ratio of the differential signal coming out of Port 2 to the common signal going into Port 2.

SCD Quadrant (Magenta):

• SCD₁₁ is the ratio of the common signal coming out of Port 1 to the differential signal going into Port 1.
• SCD₁₂ is the ratio of the common signal coming out of Port 1 to the differential signal going into Port 2.
• SCD₂₁ is the ratio of the common signal coming out of Port 2 to the differential signal going into Port 1.
• SCD₂₂ is the ratio of the common signal coming out of Port 2 to the differential signal going into Port 2.

SCC Quadrant (Green):

• SCC₁₁ is the ratio of the common signal coming out of Port 1 to the common signal going into Port 1.
• SCC₁₂ is the ratio of the common signal coming out of Port 1 to the common signal going into Port 2.
• SCC₂₁ is the ratio of the common signal coming out of Port 2 to the common signal going into Port 1.
• SCC₂₂ is the ratio of the common signal coming out of Port 2 to the common signal going into Port 2.

Single-ended S-parameters, with port order shown in Figure 4, can be mathematically converted into mixed-mode S-parameters using equations shown in Table 1.

Alternatively, Keysight ADS can simplify this process using equations on 4-Port single-ended or using 4-port Balun components, as shown in Figure 5.

Figure 5 Keysight ADS schematic used to convert from 4-Port single-ended to 2-Port mixed-mode S-parameters using equations or 4-Port Balun components. Differential and common port numbering as D1, D2, C1, C2 respectively.

Figure 6 plots mixed-mode S-parameters from equations in Table 1. Each quadrant is color coded to coincide with the respective table quadrants.

Figure 6 An example of 4-Port S-parameter mixed-mode plots of a differential transmission line.

References:

[1] M. Resso, E. Bogatin, “Signal Integrity Characterization Techniques”, International Engineering Consortium, 300 West Adams Street, Suite 1210, Chicago, Illinois 60606-5114, USA, ISBN: 978-1-931695-93-0
https://www.amazon.com/Signal-Integrity-Characterization-Techniques-Bogatin-ebook/dp/B07P9277WY/ref=sr_1_fkmr0_1?keywords=bogaitn+resso&qid=1581289220&sr=8-1-fkmr0

[2] A. Huynh, M. Karlsson, S. Gong (2010). Mixed-Mode S-Parameters and Conversion Techniques, Advanced Microwave Circuits and Systems, Vitaliy Zhurbenko (Ed.), ISBN: 978-953-307-087-2,InTech, Available from: http://www.intechopen.com/books/advanced-microwave-circuits-and-systems/mixed-mode-s-parameters-and-conversion-techniques.

[3] Alfred P. Neves, Mike Resso, and Chun-Ting Wang Lee, “S-parameters: Signal Integrity Analysis in the Blink of an Eye”, Signal Integrity Journal, https://www.signalintegrityjournal.com/articles/432-s-parameters-signal-integrity-analysis-in-the-blink-of-an-eye

Keysight Advanced Design System (ADS) [computer software], (Version 2020). URL: http://www.keysight.com/en/pc-1297113/advanced-design-system-ads?cc=US&lc=eng.