S-parameters, or scattering parameters, have become the de facto standard for describing the electrical properties of interconnects. They are one of the few concepts that bridge the worlds of the microwave designer, who typically lives in the frequency domain, and the digital designer, who typically lives in the time domain.

Everything you ever wanted to know about the electrical properties of an interconnect—a connector, a scope probe, a circuit-board trace, a circuit-board via, or a cable—is contained in the interconnect’s S-parameters. But you need to use a consistent method for assigning the port index labels to inputs and outputs or you risk obtaining misleading S-parameter values, which will lead to incorrect interpretations.

Regardless of whether S-parameters come from measurements, circuit simulations, or electromagnetic simulations, the same formalism applies and the S-parameters behave the same. S-parameters 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 S-parameters uses a defined scheme to assign port designations, and you need to be consistent with that scheme. If you create S-parameter datafiles based on one port-labeling scheme and use a data file that assumes a different labeling, the interpretation of the S-parameters and the results obtained using them will be wrong. This very basic issue of port assignment causes the most common problem when using S-parameter 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 S-parameter model and immediately determine if it assumed the incorrect port assignment.

Each S-parameter is the ratio of the wave coming out of a port to the wave going into a port (**Figure 1**). The formalism of S-parameters describes the combination of sine waves scattered from the ports of an interconnect. Every combination of this input-output port ratio makes up an S-parameter’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 two-port interconnect such as a PCB (printed-circuit 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 S-parameter matrix element corresponding to a wave that goes into port 1 and reflects back out of port 1 is labeled as S11. For historical reasons, S11 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 S-parameter 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 S-parameters is the order of the index numbers used to label each S-parameter matrix element. If a signal were to go into port 1 and come out port 2, you might assume its label would be “S12.” 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 S-parameter 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, S11, 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 through-interconnect 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 50-Ω 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.

Now comes the confusing part. If there are multiple interconnects, such as two adjacent transmission lines on a circuit board, there are two equivalent ways of labeling the port index numbers (**Figure 3**). In case 1, the opposite ends of one line are labeled port 1 and port 2, and the opposite ends of the other line are labeled port 3 and port 4. In this labeling scheme, the insertion loss of one line is still the S_{21} matrix element.

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 near-end crosstalk is S_{21}.

Both labeling approaches are legal and used in the industry. Both ways are correct. The interpretation of the same-labeled S-parameter 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 near-end 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 near-end crosstalk. These S-parameters are just as valid and just as well-defined as when labeled with the index port assignment of case 1. But if you use the S-parameter 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 S-parameter 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 S-parameter measurement used the second case as its port assignment. The S_{21} matrix element, looking like near-end crosstalk, is confirmation.

Knowing which port assignment was used is critical for two reasons. The end user of the model usually connects the S-parameter 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 single-ended 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 single-ended S-parameter matrix elements. If you assume the incorrect port assignments when calculating the differential S-parameters, the resulting differential S-parameters will be wrong.

To illustrate this problem, we measured the S-parameters from two stripline traces and stored them in a fourport S-parameter matrix using the case 1 port-labeling scheme. We then calculated the differential S-parameters 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.

Unfortunately, S-parameter 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 S-parameters from numerous sources, different files could have been created with different labeling schemes. This mix-up in the labeling scheme for the ports is the number-one source of confusion and the root cause of wrong results when using Sparameter models. (S-parameters 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 two-port 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.

Regardless of which labeling approach you use, the first thing you should do when you get a new S-parameter data file is look at the S_{21} and S_{31} terms. If S_{21} looks like an insertion loss, you know the case 1 port-labeling scheme was used. If S_{31} looks like an insertion loss, you know the case 2 port assignment was used. If it is case 2, you can ask the engineer who created the S-parameter file to consider using a less-confusing port assignment in the future.

**Eric Bogatin** is a signal-integrity 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 signal-integrity 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 high-energy physicists and electrical engineers. alan. blankman@lecroy.com.