### Introduction

S-parameters are commonly used in time-domain analysis in signal integrity. Many times s-parameter measurements are made in ignorance of the timedomain implications leading to incorrect performance in simulation and leaving engineers scratching their heads wondering what went wrong. This paper will help dispel any confusion regarding s-parameters in the time domain and in the end provide guidance that is applicable not only to time-domain reflectometer (TDR) based s-parameter measurements, as provided by the WavePulser 40iX, but to measurements made with a vector network analyzer (VNA), as well.

### Time-domain Implications of S-Parameters

The various parameters associated with s-parameters are shown in table 1. The names of the inter-linked parameters are shown on the left, with the variable names in the second column. The table is broken into a top section containing the commonly understood frequency-domain implications and a bottom section containing the less well understood time-domain implications.

The third column contains what are referred to as the microwave engineer equations, although most microwave engineers would consider only the first three variables and equations that pertain to the frequency domain. Usually, the end frequency is known and the desired frequency resolution is known somehow, and all that is necessary is to determine the number of points required for the measurement. The time-domain equations in the bottom section are grayed because usually the microwave engineer does not consider this aspect.

The last column contains what are referred to as the signal integrity equations. In these equations, there are two base variables assumed to affect all the others. These are the end frequency, the highest frequency of interest, and the impulse response length. If you are unfamiliar with the concept of impulse response length, simply consider it as twice the amount of time it takes for the response to an applied ideal impulse to completely die out1. Later, this paper discusses how to estimate that amount and how to measure it.

### Impulse Response Length and Time Aliasing

Most readers will be familiar with the topic of frequency aliasing in time-sampled systems. Time aliasing is the analogous behavior in frequency sampled systems, as s-parameter measurements are.

The impulse response of a system, as far as waves are concerned, can be calculated as the inverse discrete Fourier transform (IDFT) of each s-parameter. Although we like to think of this as the impulse response, it is actually the impulse train response [1], meaning that it is actually the response of the system to a train, or sequence of impulses applied at a period of L. Since the impulse train is assumed to extend to infinity, all the responses are the same, but if L is shorter than the actual impulse response length of the system, then portions of the impulse response calculated using the IDFT will be from impulses that occurred earlier and will be in the wrong location (hence the words “time aliasing”). Therefore, in dealing with s-parameters in signal integrity, where the time-domain implications are important, the impulse response length is one of the most important parameters. It must be known prior to taking your s-parameter measurements, and as seen in table 1, all other variables are derived from the end frequency and impulse response length.

### Impulse Response Length and Electrical Length

In microwaves, the electrical length of a device refers to the phase at a particular frequency. In signal integrity, one prefers to think of electrical length as the amount of time required for a wave to propagate through a device from one end to the other. Although one might know the electrical length of a device, there is no exact and simple relationship between the electrical length and the impulse response length. That being said, there are some relatively simple rules of thumb, and these can be tested easily (at least with a TDR instrument).

For a perfectly matched (to the reference impedance) and lossless device, the impulse response length is exactly twice the electrical length, but one likes to think that it is at least 4 the electrical length for return loss measurements, because the wave must propagate to the end of the device and back in order to determine that it is perfectly matched. This, however, is an ideal situation, which is esoteric, because no matter the reference impedance, the device will not be perfectly matched to the measurement instrument.

The recommended rule of thumb to use is based on the number of times the launched wave bounces around in a device before it can be considered to have died out. Defining a value M as an integer multiplier, for return loss, M = 2 means that the wave goes down and back once, M = 4 means it goes down and back twice, etc. Thus, L = 2 M  EL, where EL is the electrical length.1 For the thru response, one more trip is added, and M = 1 means that the wave simply gets to the end, M = 3 means that the wave goes down and back and back again, M = 5 means it goes down and back twice and then back again, etc. The recommendation for the thru response is to consider M = 5 for most devices, M = 3 for very long and lossy devices, and M = 11 for nearly lossless, but very, very poorly matched devices. For a single-port device, M = 4 is recommended.

In some cases, a device does not have actual length, but has a long impulse response. An example of such a case is a direct current (DC) blocking capacitor.

In all cases, the resulting s-parameters, whether calculated or measured, should be examined in the time domain, looking at both the corresponding step and impulse response. Both should be examined for settling and causality issues. If these are found, the s-parameters will definitely not function properly in time-domain simulations and

the calculation or measurement must be repeated with higher frequency resolution. Fortunately, the WavePulser product allows recalculation of the s-parameters from the original TDR acquisitions.

An example of incorrect and correct frequency resolution, and therefore impulse response length, is shown in figure 1 and figure 2, respectively. The frequency resolution difference between the incorrectly and correctly sampled magnitude responses in figure 1a and 2a are barely discernible, and looking in the frequency domain only does not reveal any problems. However, when viewed in the time domain [2], the impulse and step response plots in figure 1b and figure 1c reveal causality violations (shown in red). It is clear that this is due to the end of the time-domain responses wrapping into the negative time locations due to insufficient length. These violations are not actually too bad, but there are many worse scenarios. The impulse and step response plots in figure 2b and figure 2c are shown to be well settled due to the finer frequency resolution (longer impulse response length).

### Pulser Repetition Rate and Acquisition Length

When taking measurements with the WavePulser 40iX, one must consider the pulser repetition rate. The entire impulse response of the device under test (DUT) must fit inside a single period, otherwise time aliasing will result. Unfortunately, a complication is added in that this impulse response must consider the length of the path between each pulser/sampler, including all cables and fixtures, in addition to the length of the DUT. Since all cables and fixturing, in addition to the DUT, provide some impedance discontinuities, one must consider the fact that any wave launched into the system will reflect off of these interfaces, causing the actual impulse response length to be longer than the electrical length of the path.

In figure 3, three path considerations are shown. For return loss measurements as shown in 3a, the minimum

that is required is that wave propagates to the end of the path and back one time (P = 1). Even better would be to acquire two full path transits (P = 2), as there is a small amount of wave that makes it through this path. Virtually none of the launched wave makes it back and forth three times (P = 3).

In the case of Insertion loss measurements, as shown in 3b, we know that the wave must get from one pulser/sampler to the other (P = 1), but the measurement requirement is at least three path transits (P = 2). Generally, very little energy makes it five times through (P = 3).

There are other paths possible as waves bounce back and forth in the system, but usually these are insignificant sources of impulse response length, as the path transits described are already quite long.

Using the fact that the time from the internal pulser/sampler to the end of the cable for a port is 3.75 ns and that 5 ns is trimmed from each acquisition to account for the time just before the pulse, equations are developed and maximum electrical lengths tabulated for each of the acquisition length modes in the WavePulser, and for each of the path assumptions for the impulse response. These are shown in table 2. The equations are provided in table 2d. Multi-port (more than one) values for insertion loss measurements are provided in table 2a, where the P = 2 (highlighted in green) values are adequate, with P = 3 (highlighted in light green) providing maximum performance. Similarly, multi-port values for return loss measurements are provided in table 2b, where the P = 1 values are adequate, with P = 2 providing maximum performance. When only a single-port return loss measurement is made, one need only account for the fixture and cabling of a single port and the DUT, so electrically longer devices can be measured, as shown in table 2c.

Note that these electrical lengths refer to everything connected to the cable ends of the WavePulser, and must include any fixtures or adapters, even if these are to be de-embedded from the final measurement.

In TDR usually only one edge is fast, with the other edge being considered dirty with a poor impedance match [3]. Even though the WavePulser uses an impulse, the acquisition length modes correspond to one half of the pulser period. This means that even if there are very tiny reflections after the impulse response length is considered sufficiently died down, these are simply removed from the measurement and do not cause time aliasing problems.

The conclusion is that in the default 50 ns acquisition length, devices up to 1.5 ns in length are perfectly measured, but it also works well for devices up to 7.5 ns. This probably encompasses 90% of testing needs, with the longer modes providing measurements of devices over 200 ns of electrical length.

### Direct Measurement of Impulse Response Length

All the information provided here pertains to the VNA, as well, but the details are different. A VNA it must be set up to have adequate dwell time at each frequency point. Most VNA users don’t know about this and for the most part, don’t get burned by this because the default dwell time is adequate for most device lengths, just as the 50 ns acquisition length mode is adequate in the WavePulser.

That being said, it is very easy with a TDR instrument to inspect things and ascertain the proper settings by simply looking at the raw TDR traces. This is done by accessing the TDR/TDT tab on the WavePulser menu, as shown in figure 4. Here, the four instrument ports and their setup information are shown on the left with the color of the waveform, whether the trace is being displayed, and whether the pulser is on. This example is is set up for a two-port measurement with TDR port 1 sending pulses. In the middle of the menu are the acquisition controls, which are set to ten averages, in step display mode, and with SinX/X enabled. Finally, the right side of the menu controls the zooming.

Pressing the Acquire button causes the TDR/TDT traces to be acquired and displayed. By zooming in on the top of the TDR and TDT step response, you can determine the required acquisition length.

An example is shown in figure 5 for an 18-inch cable using the WavePulser. The TDR and TDT waveforms are shown in figure 5a and are shown zoomed vertically in figure 5b. The TDR waveforms are the most interesting. In figure 5b, because the path from the pulser/sampler to the end of the cable is 3.75 ns, one sees the impedance discontinuities inside the unit up to about 3 ns and the end of the cable at 7.5 ns. The DUT is located between 7.5 and 10.5 ns, after which one sees the reflections from port 2 of the WavePulser. If one zooms these traces

extremely, one can see that the reflections continue. To help visualize this, the absolute magnitude of the pulser waveform in impulse display mode (not the step display mode) is shown in figures 5c and 5d. In 5c, one sees that the reflection from port 2 of the WavePulser gets back to port 1 at 18 ns. The secondary reflection (the reflection that went down and back twice) ends at 36 ns, but it is 40 ns down, or 1% the size of the first reflection, and is about 80 dB below the incident pulse size. This is shown similarly in 5d where the incident waveform arrives at port 2 at 9 ns, and the secondary reflection finishes at 93 = 27 ns. The tertiary reflection ends theoretically at 95 = 45 ns, the limit in the 50 ns acquisition length mode, about 100 dB below the first reflection.

From this example, it becomes clear why the recommendation for good measurements is to consider two transits (down and back) for return loss measurements and three for insertion loss measurements of the entire system containing the DUT. Four transits for return loss and five for insertion loss would capture absolutely the entire signal, but this is generally considered overkill.

### WavePulser Setup and Impulse Response Limiting

The frequency settings for s-parameter measurements are in the Setup menu for the WavePulser, as shown in figure 6. Aside from the number of ports and the averaging mode to use, the frequency settings are all that is needed to make a measurement. To maintain compatibility with the VNA, the instrument provides the similar settings of end frequency and number of points. Since the instrument measures to DC, unlike the VNA, and because DC is always desired in signal integrity measurements, the actual number of points is one more than this number. To help see this, the Delta Frequency is calculated and shown below along with the Time Length, which is the positive portion of the impulse response length associated with the s-parameters. Finally, the Acquisition Length is provided, which determines the length of TDR measurements shown, which is one-half of the reciprocal of the pulser repetition rate.

TheWavePulser software rounds the numbers entered to nice numbers to avoid length resampling of the s-parameters and limits the values allowed to 8,000 points. This can be overridden by enabling Fine Mode, which allows up to 40,000 points (500 s time length, or 1 s impulse response length, needed at the maximum acquisition length mode).

Below this are various physicality enforcements, including passivity, reciprocity, and causality, along with impulse response limiting. Impulse response limiting means that after the s-parameters are calculated, the impulse response is zeroed out beyond plus/minus this amount. The simple reason one might want to limit the impulse response length is the smooth out and clean up the final s-parameter measurement, so as not to include areas of the impulse response beyond that which is physically possible. For example, one might calculate 4; 000 points to 40 GHz, which according to table 1, has a positive time length of 50 ns, and an impulse response length of 100 ns on a device that has an electrical length of 500 ps. Using the rules of thumb provided earlier, a device like this would never require more than L = 2  0:5  5 = 5 ns, so it’s reasonable to limit the impulse response length to 2:5 ns. When this is done, one observes generally cleaner measurements.

Upon reading this, one might wonder why use such a fine frequency resolution and then limit the response. Why not use 5 ns (or f = 200MHz, which is 200 points to 40 GHz) to start with? This topic is quite Teledyne LeCroy WavePulser 40iX Pulser Repetition Rate and Frequency Resolution page | 7 of 8 complicated, so only relatively simple statements will be made here. First, since all the calibrations are made in the frequency domain, although there is time aliasing going on, the calculations work out properly in the end, even with 200 points. But, if there are any time-domain de-embedding operations utilized, like time gating or impedance peeling, these operations must be performed with the proper sense of time, meaning the s-parameters calculated up to that de-embedding step must have sufficient impulse response length. Furthermore, although the frequency-domain calibrations and de-embedding all work out, even in the presence of time aliasing, if and when small errors crop up, these errors will be time aliased if low frequency resolution is utilized. Therefore, certainly when debugging calibration or de-embedding issues, it makes sense to measure the s-parameters with sufficient resolution to encompass the combination of the DUT and fixtures, examine the de-embedded sparameters for causality issues caused by improper de-embedding, and then limit the impulse response length to remove any small errors.

### Conclusion

S-parameter frequency resolution and its relationship to impulse response length is a complicated topic, but a topic that must be understood and handled properly by the signal integrity engineer, whether using a VNA or TDR instrument to measure s-parameters. Since time-domain implications of s-parameters used in signal integrity analysis are so important, one should understand the electrical length limitations of the measurement instruments and the controls such as the acquisition length mode of the WavePulser 40iX. That being said, the default acquisition length of 50 ns is adequate for most device measurements, with the longest mode supporting the measurement of devices up to 200 ns in electrical length. More details on these topics can be found in the references.