# S-Parameters for digital designers: S-Parameter data can provide valuable information about the characteristics of a PCB interconnect.

As multi-gigabit interfaces, like PCI Express, have continued to
advance in frequency for each generation (PCI Expression Generation 2
operates at 5 Gbps), digital designers are being pushed into the RF
realm of microwave theory. Once only reserved in the toolbox of RF
engineers, S-Parameters are now being widely used on high-speed designs
to analyze the passive interconnects in a system. S-Parameters are ideal
for digital designers because they can be used as the behavioral model
of a network, translated into terms of reflection and loss.

For passive interconnect structures, like transmission lines and vias, as the frequency increases, so does the dependency on loss. Standard methods of analyzing and modeling these types of structures are based upon certain assumptions of Maxwell's equations that no longer hold true when considering a wide range of frequency dependent phenomena.

The advantage of using S-Parameters is that they describe how the structure behaves in the frequency domain, since they contain data taken across a wide frequency range. Understanding how to properly generate S-Parameter data and use the data to represent a structure in simulations will provide designers with a new way to tackle high-speed designs.

Basic S-Parameter Theory

While using an impedance matrix to define a network might be more intuitive, it is actually more difficult to get the proper measurements; therefore, S-Parameters are the choice of the digital designer. S-Parameters are based on network theory that states a linear system can be characterized by input and output ports without regard to the actual system. The S-Parameters can be used to describe the electrical characteristics of any passive interconnect in a system, over a wide bandwidth to account for frequency dependencies. S-Parameters characterize a network by measuring the transmitted and reflected power traveling waves at the defined ports. (1) A network can have any number of ports, but the focus here will be on a simple 2-port network, as shown in FIGURE 1. Linear equations can be used to describe the network in terms of injected and transmitted power waves, and once you have these liner equations, you can easily translate them to the time-domain, as well as convert them to other formats such as Y- and Z-Parameters.

[S.sub.11] = ratio of reflected to incident wave = Return Loss

[S.sub.21] = ratio of transmitted to incident wave = Insertion Loss

The Return Loss is sometimes referred to as the complex reflection coefficient. Digital designers are familiar with reflection coefficients (from reflections of transmission lines in the time domain), and the Return Loss can be viewed as the reflection coefficient of the interconnect in the frequency domain. Return Loss represents how much of the signal is reflected back due to any impedance mismatch.

The Insertion Loss represents how much of the signal is lost from transferring energy from one port to another; the loss is due to impedance discontinuities and resonance effects of the interconnect.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Understanding Return Loss and Insertion Loss

One of the more challenging aspects of dealing with S-Parameters is understanding how the Return Loss and Insertion Loss parameters relate back to the interconnect performance. We'll look at examples of each parameter to understand them better.

We have already discussed how the Return Loss is really about understanding the impedance of the interconnect by representing the reflection coefficients. The magnitude of the Return Loss can be displayed on a rectangular X-Y plot, either in linear or log format across frequency (FIGURE 2). Since S-Parameters are frequency models, they will often be plotted on log format in decibels dB where:

dB = -20log(p) and

[rho] = | reflection coefficient I

To make sense of the Return Loss data, we can apply a useful metric to the plot and state that we want our reflection coefficient to be less than 10% for our interconnect because that satisfies our noise margin budget for this interconnect. By looking at the Return Loss graph, we can see across frequency, where the interconnect is below -20 dB (a 10% reflection coefficient is [rho]=0.01 and dB=-20). The smaller the reflection coefficient, the larger the Return Loss should be. If you know the bandwidth of the interconnect, the Return Loss data can be used to quickly decide if the interconnect impedance will be adequate across the desired frequency range.

The Return Loss plot shows the reflection coefficient of the interconnect at different frequencies, the nulls seen in the plot are resonances due to impedance mismatches. The Return Loss can be used to calculate the impedance (determined by the magnitude) and the electrical delay (determined by the phase) of the interconnect. Useful information about the interconnect can be derived from the Return Loss data. The S-Parameter data can also be easily translated into different formats, such as Admittance Y-Parameters that may be more intuitive when talking about impedance and delay. (2)

[FIGURE 3 OMITTED]

The Insertion Loss represents how much of the signal is degraded going through the interconnect. The magnitude of the Insertion Loss can be displayed on a rectangular X-Y plot, either in linear or log format across the frequency, as shown in FIGURE 3. We will talk about Insertion Loss in dB, since that is the more popular format. A perfect impedance matched interconnect will have a loss of 0 dB. Remember that Insertion Loss is the ratio of how much energy is transmitted, so if all the energy is transmitted, the ratio comes out to be a magnitude of 1 because there is no loss. (A magnitude of 1 in dB is 0 dB) To make sense of the Insertion Loss data, we can apply a useful metric to the plot and state that we want at least 70% of the signal to get transmitted through the interconnect to satisfy our noise budget for this interconnect. By looking at the graph, we can see across frequency, where the Insertion Loss is smaller than roughly -3 dB. (The 3 dB metric point is as common as the 50 Ohm transmission line across disciplines). Typically, you will see lower loss at low frequencies, and as the frequency increases, so does the attenuation and loss.

So far, we have only talked about the magnitudes of S-Parameters, but the phase can be useful as well. An ideal interconnect will have the same delay across all frequencies, but this isn't always the case, due to frequency dependencies in the material properties of interconnects. The Group Delay is the derivative of the phase shifts of S21 across frequency. What the Group Delay really detects is if the interconnect delay changes across frequency. When looking at a Group Delay plot, it should be flat across frequency; if not, it can show how much the delay shifts with frequency.

Quality Checks for S-Parameter Models

In order to understand an S-Parameter model better and do some basic quality checks, some type of S-Parameter data viewer is highly recommended, such as Mentor Graphics' HyperLynx TouchStone Viewer, shown in FIGURE 4. Most commercial EDA tools, such as Synopsys Waveview Analyzer, also provide support for viewing and graphing S-Parameter model data.

[FIGURE 4 OMITTED]

The first quality check to perform on an S-Parameter model is a causality check. (3) The propagation delay of an S-Parameter model cannot be faster than the electromagnetic equations governing the material properties of the interconnect. Non-causal S-Parameter models will exhibit incorrect propagation delays for the interconnect structure being represented. Think of a microstrip transmission line that has a propagation of 10 ps/in., instead of a more reasonable 150 ps/in, for FR-4 dielectric material. The best way to check for causality issues is to use a Polar Plot, a graph of the imaginary versus the real (FIGURE 5); a Smith Chart, a special case Polar Plot, can also be used. When viewing the S-Parameter data on a Polar Plot, the data should have a clockwise rotation, indicating a positive Group Delay. The resonances create the circles in the path. The data should also start and end on the real axis and be smooth with a lot of resolution.

The second quality check to perform is a passivity check. A passive system does not generate energy, so there should be no energy generated between ports. A passivity check will look at the sum of the energy input to the network versus the sum of the energy output from the network, and if there is energy generated, an error is reported. The passivity check is performed on the entire S-Parameter model using all the ports, and most EDA tools can check the passivity for you and report any errors.

S-Parameter Quality Review Checklist:

* [S.sub.11] and [S.sub.12] magnitude < 1 (otherwise there would be negative impedance)

* Poor Symmetry ([S.sub.12] [not equal to] [S.sub.21])

* Polar Plot [right arrow] clockwise rotation over frequency

* Polar Plot [right arrow] smooth curve over frequency

* Polar Plot [right arrow] enough data points

* Passivity Check (available in most simulation tools)

Many S-Parameter models need some editing before using in a simulation to smooth out data or correct minor passivity issues. Different tools will allow you to view both the original and the modified S-Parameter model. There is a work around for some tools. It involved running simulations using the original versus the modified S-Parameter model. These are completed using different test benches to validate that the changes are in line with what was expected.

[FIGURE 5 OMITTED]

Tips on Generating S-Parameter Models

To avoid passivity and causality errors when generating S-Parameter models from either lab measurement or simulation, the following guidelines (4) should be adhered to:

* A broad frequency range sweep should be used (good rule of thumb: 5th harmonic of fundamental frequency).

* A "close to DC" point needs to be included. This is easier to do in simulation than in a lab measurement. Care needs to be taken to get as much resolution as possible, at the lowest frequency, when manually editing data to add the DC point.

* A dense sampling of data points is necessary. Ideally, an adaptive type of sampling would be used to generate a step size of data points rather than a "one fits all" approach. Linear spacing may be needed depending on your simulation tool or use of model.

* Provide high resolution with lots of data points. File size can be an issue, but whether an adaptive approach is used or not, there needs to be a lot of data points.

* Measurement noise needs to be calibrated out or smoothed out in software.

Simulating S-Parameters in the Time Domain

While useful characteristics of the interconnect, like impedance, can be derived from the S-Parameters most of the time, the goal is to use the S-Parameter model in a time domain simulation to validate the interface. Since S-Parameters contain frequency-based data, the simulation tool needs to convert this data into the time domain. The most common approach is to do so using an Inverse Fast Fouirier Transform (IFFT) and then use convolution to generate the transient response. This approach has some disadvantages, though. It takes a lot of simulation time to do this conversion from the frequency to the time domain. Also, by using a direct convolution method, passivity and causality issues are sometimes either ignored or incorporated into the time domain result, causing simulation convergence issues. Some simulation tools require a linear spacing of the S-Parameter data to perform the IFFT, which with linear spaced data can lead to insufficient data points in the model.

There are many different schools of thought on better methods to use S-Parameter data in time domain simulations. Many commercial EDA simulators offer different algorithm methods to simulate the data in the time domain. A popular method involves representing the S-Parameter data with poles-residues based on a curve that fits the data. (5) Since only stable poles are used, it can eliminate causality errors. This method also leads to efficient time domain simulation, since the poles-residues are represented with non-frequency based functions or elements so no convolution is required.

It is worthwhile for the digital designer to talk with the simulation vendor in order to understand the methods available for running efficient transient simulations. While there are different methods available to speed up simulation times and reduce convergence issues, there is always a trade-off in accuracy and performance, so the right balance has to be found.

Conclusion

S-Parameters provide a new analysis tool for digital designers to add to their signal integrity toolbox. The S-Parameter data can be viewed and analyzed to obtain valuable information about the characteristics of the PCB interconnect. If care is taken when generating S-Parameter data from measurement or simulation, headaches can be avoided later on. By understanding how the S-Parameter data is being used in time domain simulations, digital designers can ask the right questions to speed up simulations and avoid convergence issues. As technologies keep advancing, the S-Parameter format will continue to gain importance for all engineers.

REFERENCES

(1.) Test and Measurement, Application Note 95-1, Hewlett Packard.

(2.) Digital Communications Test and Measurement, Dennis Derickson and Marcus Muller, Prentice Hall Signal Integrity Library, December 2007.

(3.) P. Triverio, S. Grivet-Talocia, M.S. Nakhla, F.G. Canavero, R. Achar, "Stability, causality, and passivity in electrical interconnect models" IEEE Trans. on Adv. Pack., v. 30, N4, p. 795-808.

(4.) "Rational Function Computing with Poles and Residues," Richard J. Fateman, Computer Science Division, University of California, Berkeley, March 21, 2006.

(5.) S-Parameter Model Quality, "How to Check and Improve S-Parameter Data" Vladimir Dmitriev-Zdorov and Steve Kaufer, AppNote 10029, Mentor Graphics, 2006.

TIMOTHY COYLE is principal consultant and owner at Signal Consulting Group LLC; tim.coyle@siconsultant.com.

For passive interconnect structures, like transmission lines and vias, as the frequency increases, so does the dependency on loss. Standard methods of analyzing and modeling these types of structures are based upon certain assumptions of Maxwell's equations that no longer hold true when considering a wide range of frequency dependent phenomena.

The advantage of using S-Parameters is that they describe how the structure behaves in the frequency domain, since they contain data taken across a wide frequency range. Understanding how to properly generate S-Parameter data and use the data to represent a structure in simulations will provide designers with a new way to tackle high-speed designs.

Basic S-Parameter Theory

While using an impedance matrix to define a network might be more intuitive, it is actually more difficult to get the proper measurements; therefore, S-Parameters are the choice of the digital designer. S-Parameters are based on network theory that states a linear system can be characterized by input and output ports without regard to the actual system. The S-Parameters can be used to describe the electrical characteristics of any passive interconnect in a system, over a wide bandwidth to account for frequency dependencies. S-Parameters characterize a network by measuring the transmitted and reflected power traveling waves at the defined ports. (1) A network can have any number of ports, but the focus here will be on a simple 2-port network, as shown in FIGURE 1. Linear equations can be used to describe the network in terms of injected and transmitted power waves, and once you have these liner equations, you can easily translate them to the time-domain, as well as convert them to other formats such as Y- and Z-Parameters.

[S.sub.11] = ratio of reflected to incident wave = Return Loss

[S.sub.21] = ratio of transmitted to incident wave = Insertion Loss

The Return Loss is sometimes referred to as the complex reflection coefficient. Digital designers are familiar with reflection coefficients (from reflections of transmission lines in the time domain), and the Return Loss can be viewed as the reflection coefficient of the interconnect in the frequency domain. Return Loss represents how much of the signal is reflected back due to any impedance mismatch.

The Insertion Loss represents how much of the signal is lost from transferring energy from one port to another; the loss is due to impedance discontinuities and resonance effects of the interconnect.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Understanding Return Loss and Insertion Loss

One of the more challenging aspects of dealing with S-Parameters is understanding how the Return Loss and Insertion Loss parameters relate back to the interconnect performance. We'll look at examples of each parameter to understand them better.

We have already discussed how the Return Loss is really about understanding the impedance of the interconnect by representing the reflection coefficients. The magnitude of the Return Loss can be displayed on a rectangular X-Y plot, either in linear or log format across frequency (FIGURE 2). Since S-Parameters are frequency models, they will often be plotted on log format in decibels dB where:

dB = -20log(p) and

[rho] = | reflection coefficient I

To make sense of the Return Loss data, we can apply a useful metric to the plot and state that we want our reflection coefficient to be less than 10% for our interconnect because that satisfies our noise margin budget for this interconnect. By looking at the Return Loss graph, we can see across frequency, where the interconnect is below -20 dB (a 10% reflection coefficient is [rho]=0.01 and dB=-20). The smaller the reflection coefficient, the larger the Return Loss should be. If you know the bandwidth of the interconnect, the Return Loss data can be used to quickly decide if the interconnect impedance will be adequate across the desired frequency range.

The Return Loss plot shows the reflection coefficient of the interconnect at different frequencies, the nulls seen in the plot are resonances due to impedance mismatches. The Return Loss can be used to calculate the impedance (determined by the magnitude) and the electrical delay (determined by the phase) of the interconnect. Useful information about the interconnect can be derived from the Return Loss data. The S-Parameter data can also be easily translated into different formats, such as Admittance Y-Parameters that may be more intuitive when talking about impedance and delay. (2)

[FIGURE 3 OMITTED]

The Insertion Loss represents how much of the signal is degraded going through the interconnect. The magnitude of the Insertion Loss can be displayed on a rectangular X-Y plot, either in linear or log format across the frequency, as shown in FIGURE 3. We will talk about Insertion Loss in dB, since that is the more popular format. A perfect impedance matched interconnect will have a loss of 0 dB. Remember that Insertion Loss is the ratio of how much energy is transmitted, so if all the energy is transmitted, the ratio comes out to be a magnitude of 1 because there is no loss. (A magnitude of 1 in dB is 0 dB) To make sense of the Insertion Loss data, we can apply a useful metric to the plot and state that we want at least 70% of the signal to get transmitted through the interconnect to satisfy our noise budget for this interconnect. By looking at the graph, we can see across frequency, where the Insertion Loss is smaller than roughly -3 dB. (The 3 dB metric point is as common as the 50 Ohm transmission line across disciplines). Typically, you will see lower loss at low frequencies, and as the frequency increases, so does the attenuation and loss.

So far, we have only talked about the magnitudes of S-Parameters, but the phase can be useful as well. An ideal interconnect will have the same delay across all frequencies, but this isn't always the case, due to frequency dependencies in the material properties of interconnects. The Group Delay is the derivative of the phase shifts of S21 across frequency. What the Group Delay really detects is if the interconnect delay changes across frequency. When looking at a Group Delay plot, it should be flat across frequency; if not, it can show how much the delay shifts with frequency.

Quality Checks for S-Parameter Models

In order to understand an S-Parameter model better and do some basic quality checks, some type of S-Parameter data viewer is highly recommended, such as Mentor Graphics' HyperLynx TouchStone Viewer, shown in FIGURE 4. Most commercial EDA tools, such as Synopsys Waveview Analyzer, also provide support for viewing and graphing S-Parameter model data.

[FIGURE 4 OMITTED]

The first quality check to perform on an S-Parameter model is a causality check. (3) The propagation delay of an S-Parameter model cannot be faster than the electromagnetic equations governing the material properties of the interconnect. Non-causal S-Parameter models will exhibit incorrect propagation delays for the interconnect structure being represented. Think of a microstrip transmission line that has a propagation of 10 ps/in., instead of a more reasonable 150 ps/in, for FR-4 dielectric material. The best way to check for causality issues is to use a Polar Plot, a graph of the imaginary versus the real (FIGURE 5); a Smith Chart, a special case Polar Plot, can also be used. When viewing the S-Parameter data on a Polar Plot, the data should have a clockwise rotation, indicating a positive Group Delay. The resonances create the circles in the path. The data should also start and end on the real axis and be smooth with a lot of resolution.

The second quality check to perform is a passivity check. A passive system does not generate energy, so there should be no energy generated between ports. A passivity check will look at the sum of the energy input to the network versus the sum of the energy output from the network, and if there is energy generated, an error is reported. The passivity check is performed on the entire S-Parameter model using all the ports, and most EDA tools can check the passivity for you and report any errors.

S-Parameter Quality Review Checklist:

* [S.sub.11] and [S.sub.12] magnitude < 1 (otherwise there would be negative impedance)

* Poor Symmetry ([S.sub.12] [not equal to] [S.sub.21])

* Polar Plot [right arrow] clockwise rotation over frequency

* Polar Plot [right arrow] smooth curve over frequency

* Polar Plot [right arrow] enough data points

* Passivity Check (available in most simulation tools)

Many S-Parameter models need some editing before using in a simulation to smooth out data or correct minor passivity issues. Different tools will allow you to view both the original and the modified S-Parameter model. There is a work around for some tools. It involved running simulations using the original versus the modified S-Parameter model. These are completed using different test benches to validate that the changes are in line with what was expected.

[FIGURE 5 OMITTED]

Tips on Generating S-Parameter Models

To avoid passivity and causality errors when generating S-Parameter models from either lab measurement or simulation, the following guidelines (4) should be adhered to:

* A broad frequency range sweep should be used (good rule of thumb: 5th harmonic of fundamental frequency).

* A "close to DC" point needs to be included. This is easier to do in simulation than in a lab measurement. Care needs to be taken to get as much resolution as possible, at the lowest frequency, when manually editing data to add the DC point.

* A dense sampling of data points is necessary. Ideally, an adaptive type of sampling would be used to generate a step size of data points rather than a "one fits all" approach. Linear spacing may be needed depending on your simulation tool or use of model.

* Provide high resolution with lots of data points. File size can be an issue, but whether an adaptive approach is used or not, there needs to be a lot of data points.

* Measurement noise needs to be calibrated out or smoothed out in software.

Simulating S-Parameters in the Time Domain

While useful characteristics of the interconnect, like impedance, can be derived from the S-Parameters most of the time, the goal is to use the S-Parameter model in a time domain simulation to validate the interface. Since S-Parameters contain frequency-based data, the simulation tool needs to convert this data into the time domain. The most common approach is to do so using an Inverse Fast Fouirier Transform (IFFT) and then use convolution to generate the transient response. This approach has some disadvantages, though. It takes a lot of simulation time to do this conversion from the frequency to the time domain. Also, by using a direct convolution method, passivity and causality issues are sometimes either ignored or incorporated into the time domain result, causing simulation convergence issues. Some simulation tools require a linear spacing of the S-Parameter data to perform the IFFT, which with linear spaced data can lead to insufficient data points in the model.

There are many different schools of thought on better methods to use S-Parameter data in time domain simulations. Many commercial EDA simulators offer different algorithm methods to simulate the data in the time domain. A popular method involves representing the S-Parameter data with poles-residues based on a curve that fits the data. (5) Since only stable poles are used, it can eliminate causality errors. This method also leads to efficient time domain simulation, since the poles-residues are represented with non-frequency based functions or elements so no convolution is required.

It is worthwhile for the digital designer to talk with the simulation vendor in order to understand the methods available for running efficient transient simulations. While there are different methods available to speed up simulation times and reduce convergence issues, there is always a trade-off in accuracy and performance, so the right balance has to be found.

Conclusion

S-Parameters provide a new analysis tool for digital designers to add to their signal integrity toolbox. The S-Parameter data can be viewed and analyzed to obtain valuable information about the characteristics of the PCB interconnect. If care is taken when generating S-Parameter data from measurement or simulation, headaches can be avoided later on. By understanding how the S-Parameter data is being used in time domain simulations, digital designers can ask the right questions to speed up simulations and avoid convergence issues. As technologies keep advancing, the S-Parameter format will continue to gain importance for all engineers.

REFERENCES

(1.) Test and Measurement, Application Note 95-1, Hewlett Packard.

(2.) Digital Communications Test and Measurement, Dennis Derickson and Marcus Muller, Prentice Hall Signal Integrity Library, December 2007.

(3.) P. Triverio, S. Grivet-Talocia, M.S. Nakhla, F.G. Canavero, R. Achar, "Stability, causality, and passivity in electrical interconnect models" IEEE Trans. on Adv. Pack., v. 30, N4, p. 795-808.

(4.) "Rational Function Computing with Poles and Residues," Richard J. Fateman, Computer Science Division, University of California, Berkeley, March 21, 2006.

(5.) S-Parameter Model Quality, "How to Check and Improve S-Parameter Data" Vladimir Dmitriev-Zdorov and Steve Kaufer, AppNote 10029, Mentor Graphics, 2006.

TIMOTHY COYLE is principal consultant and owner at Signal Consulting Group LLC; tim.coyle@siconsultant.com.

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Title Annotation: | MODELING |
---|---|

Author: | Coyle, Timothy |

Publication: | Printed Circuit Design & Fab |

Date: | May 1, 2009 |

Words: | 2271 |

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