Avoid a common Sparameter problem

Everything you ever wanted to know about the electrical properties of an interconnect—a connector, a scope probe, a circuitboard trace, a circuitboard via, or a cable—is contained in the interconnect’s Sparameters. But you need to use a consistent method for assigning the port index labels to inputs and outputs or you risk obtaining misleading Sparameter values, which will lead to incorrect interpretations.
Regardless of whether Sparameters come from measurements, circuit simulations, or electromagnetic simulations, the same formalism applies and the Sparameters behave the same. Sparameters describe how sine waves interact with and “scatter” from an interconnect. Each interconnect has “ports,” defined as the ends of the interconnect into which signals enter and from which they leave. Each port has connections to the signal conductor and its return path. Index numbers label the ports into which a signal enters and from which it scatters.
Consistency is paramount when you are labeling these ports. Software used to calculate Sparameters uses a defined scheme to assign port designations, and you need to be consistent with that scheme. If you create Sparameter datafiles based on one portlabeling scheme and use a data file that assumes a different labeling, the interpretation of the Sparameters and the results obtained using them will be wrong. This very basic issue of port assignment causes the most common problem when using Sparameter models: incorrect interpretation of the data.
By following one simple guideline, you can eliminate this problem. You will also be able to look at an Sparameter model and immediately determine if it assumed the incorrect port assignment.
Return loss and insertion loss
Each Sparameter is the ratio of the wave coming out of a port to the wave going into a port (Figure 1). The formalism of Sparameters describes the combination of sine waves scattered from the ports of an interconnect. Every combination of this inputoutput port ratio makes up an Sparameter’s matrix elements. Each matrix element is defined by the input port number (the stimulus) and the output port number (the response). This formalism applies regardless of whether the interconnect has just one port or 100 ports.

In a twoport interconnect such as a PCB (printedcircuit board) trace or a cable, there’s only one way to assign the index port labels: port 1 on one side and port 2 on the other side. The Sparameter matrix element corresponding to a wave that goes into port 1 and reflects back out of port 1 is labeled as S_{11}. For historical reasons, S_{11} is also referred to as return loss. Because impedance changes along the interconnect cause reflected waves, return loss is very sensitive to the interconnect’s impedance profile. The Sparameter corresponding to the wave going into port 1 and coming out port 2 is labeled S_{21} and is referred to, for historical reasons, as the insertion loss. It has information about reflections and is also sensitive to the losses in the interconnect.
One confusing aspect of Sparameters is the order of the index numbers used to label each Sparameter matrix element. If a signal were to go into port 1 and come out port 2, you might assume its label would be “S_{12}.” The label would be easy to remember at a glance: The signal goes into port 1 and comes out port 2.
Unfortunately, as a consequence of the matrix math formalism, the labeling scheme follows the opposite structure. The Sparameter matrix element containing information about the wave going into port 1 and coming out port 2 is actually S_{21}.
At the lowest frequency, where the physical length of the interconnect is really short compared to ¼ of a wavelength, the reflection off the front of the interconnect and the reflection from the back end of the interconnect mostly cancel out one another, so the return loss, S_{11}, is nearly zero. In decibels (dB), the return loss for a through interconnect at low frequency is almost always a large negative decibel value.
The transmitted signal, described by S_{21}, is due to the initial transmitted signal, and a small contribution from the signal reflects off port 2 to port 1, then reflects back to port 2 and, finally, out port 2. At the lowest frequency, all of the signal gets through and comes out port 2.
The insertion loss of a throughinterconnect at low frequency will be close to 0 dB.
As frequency increases, the losses in all interconnects cause the insertion loss to fall, which means a larger and more negative insertion loss in decibels. An example of the measured return and insertion loss of a typical 50Ω trace on a circuit board is shown in Figure 2.

This is an important observation: For virtually all interconnects, at the lowest frequency, you can expect the insertion loss to be nearly 0 dB. This is an easy and direct way to determine which matrix element is really the insertion loss, independent of the port labeling.
More than twoport Sparameters

We recommend that you use the case 1 labeling scheme. It’s consistent with the intuition we built up connecting insertion loss with the S_{21} matrix element, and it easily scales to more ports.
In case 2, port 1 and port 2 are the labels on the left side of the pair of lines and port 3 and port 4 are the labels on the right side of the pair. In this labeling scheme, the insertion loss of the first line is actually the S_{31} matrix element, and the nearend crosstalk is S_{21}.
Both labeling approaches are legal and used in the industry. Both ways are correct. The interpretation of the samelabeled Sparameter matrix element, however, is obviously different depending on which port assignment you use.
In the first port assignment, the insertion loss is S_{21} and you would expect it to be nearly 0 dB at low frequency. The S_{31} matrix element relates to the nearend crosstalk between the two lines and should always be very small, or a large negative decibel value at low frequency.
In the second port assignment, the insertion loss is the matrix element S_{31}. The matrix element S_{21} is the nearend crosstalk. These Sparameters are just as valid and just as welldefined as when labeled with the index port assignment of case 1. But if you use the Sparameter model created with one labeling scheme in an application that has a different labeling scheme, the result will be the same as if you had a bad model.
The way to tell which port assignment was used in an Sparameter file is to look at the S_{21} matrix element. If S_{21} looks like an insertion loss, starting out with a nearly 0 dB value at low frequency, then the port assignments were labeled as in case 1. If S_{31} looks like an insertion loss and has a nearly 0 dB value at low frequency, then the port assignments were labeled as in case 2.
As an example, Figure 4 shows the measured S_{21} and S_{31} matrix elements from a pair of stripline traces. S_{31} looks like an insertion loss, starting out at low frequency with 0 dB. This Sparameter measurement used the second case as its port assignment. The S_{21} matrix element, looking like nearend crosstalk, is confirmation.

Knowing which port assignment was used is critical for two reasons. The end user of the model usually connects the Sparameter model into a circuit by connecting circuit nodes to ports. If the port assignments are not as expected, the circuit will still simulate and you will get a resulting waveform, but it will be a completely wrong result.
In addition, it is increasingly common for two singleended transmission lines to be used as one differential pair. The differential insertion and return loss of the differential pair, designated by matrix elements SDD21 and SDD11, are created from linear combinations of the singleended Sparameter matrix elements. If you assume the incorrect port assignments when calculating the differential Sparameters, the resulting differential Sparameters will be wrong.
To illustrate this problem, we measured the Sparameters from two stripline traces and stored them in a fourport Sparameter matrix using the case 1 portlabeling scheme. We then calculated the differential Sparameters in two ways: the first correctly assumed case 1 labeling; the second incorrectly assumed case 2 labeling. Figure 5 shows the resulting differential insertion and return loss for each assumption.

An insertion loss, whether singleended or differential, will always start near 0 dB at low frequency. Clearly, the differential insertion loss assuming the wrong port assignment results in an insertion loss that is not consistent with our expectation, as it starts out with a large negative decibel value.
Recommendations for port assignments
Unfortunately, Sparameter files rarely note which labeling scheme was used to create the file, and you might forget to write down which scheme you used. If you deal with Sparameters from numerous sources, different files could have been created with different labeling schemes. This mixup in the labeling scheme for the ports is the numberone source of confusion and the root cause of wrong results when using Sparameter models. (Sparameters are confusing enough without adding another opportunity for confusion.)
To avoid this common source of confusion, we strongly recommend you adopt the habit of labeling the port index numbers with odd port numbers on the left side and even port numbers on the right. This approach has two important
advantages:
 It is consistent with the labeling of twoport interconnects. Insertion loss is still S_{21}.
 It is scalable, so for four ports, you just need to add the additional lines and continue with the labeling of 3 to 4, 5 to 6, 7 to 8, and so forth.
Eric Bogatin is a signalintegrity evangelist at Bogatin Enterprises, a LeCroy company. He holds an SB degree in physics from MIT and a PhD in physics from the University of Arizona in Tucson. He has been active in the signalintegrity industry for more than 30 years, writing articles and books and teaching classes. eric@bethesignal.com.
Alan Blankman is the technical product marketing manager for signal integrity at LeCroy. He holds a PhD in physics from the University of Pennsylvania and an MBA from the New York University Stern School of Business. He has more than 20 years of experience developing instrumentation and software for highenergy physicists and electrical engineers. alan.blankman@lecroy.com.
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