S-Parameters: New Look and Similarity

Bandwidth required for signal integrity analysis of PCB and packaging interconnects is growing with the increase of data rates. Evaluation of model accuracy requires validation with the measurements – this is a necessary element of successful design process with data rates above 10 Gbps. A systematic approach to the analysis to measurement validation was introduced in [1], [2]. Though, the last step in the process was a visual estimate of the closeness of models to measured data (use of “human visual system”). The process needs formal approach and possibility of automation. Feature Selective Validation (FSV) method [3] can be used for such purpose. However, it is rather complicated (not quite straightforward), has too many parameters and can be applied only to amplitudes of S-parameters. A formal single-number S-parameters similarity (SPS) metric is introduced in [4]. A new look at S-parameters in 3D Real-Imaginary-Frequency (RIF) space enabled application of the image recognition technique to computation of similarity between 2 sets of S-parameters. This article is just a brief introduction into how SPS is defined with some practical examples – more details are available in [4]. The idea was first published in Y. Shlepnev “Evaluation of S-Parameters Similarity with Modified Hausdorff Distance” at https://arxiv.org/abs/2105.10057 on May 20, 2021.

S-parameters are usually a set of S-matrices computed or measured at a number of frequency points. Each element of the matrix or S-parameter is a set of complex numbers at the frequency points. They can be plotted as magnitude and angle (Bode plot) as illustrated below for reflection and transmission parameters for two simple Beatty resonators:

We can observe some shift of resonances in two structures, but it is difficult to make any conclusion on the angles – they are close at lower frequencies, but then deviate from each other. The jumps can be un-wrapped, but it is not always straightforward for measured data.

Another possible way to visualize S-parameters is to use Nyquist or polar plots – simply plot real value of S-parameter vs. imaginary at each frequency point and connect the points as the frequency increases. A vector from 0 to each frequency point gives magnitude and angle of the corresponding S-parameter at that frequency. The same reflection and transmission parameters shown above on the Bode plots are shown below on the polar plots:

Such plots are even more confusing and less usable for the comparison purpose.

What if we plot the real and imaginary parts in 3D with the frequency as additional or Z-axis? We will get 3D spiral plots as shown below for the same set of S-parameters for 2 Beatty resonators:

The plots do not have intersections as the polar plots and no jumps of angles as on Bode plots. The visual comparison is still difficult, but we can apply an image recognition technique to measure the similarity of the data sets. In particular, Modified Hausdorff Distance (MHD) can be used to measure the distance between the two data sets (S-parameters of 2 Beatty resonators in this case):

S-parameters similarity (SPS) metric with span from 0 (no similarity) to 100% (identical structures) can be introduces with the MHD computed for each element of matrix and for the whole matrix as shown on the picture above. Very easy and straightforward – it is just a few lines of code in Matlab as shown in [4].

 As a practical case let’s evaluate all models with all measurements for CMP-28 validation platform [5] from Wild River Technology (https://www.wildrivertech.com/). Values of SPS are computed with Simbeor SDK for all test structures on CMP-28 validation platform and are shown in the next table for 3 different comparison bandwidths – 10 GHz, 35 GHz and 50 GHz for single-ended S-parameters (SPP_SE) and for the mixed-mode S-parameters (SPS_MM): 

“n/a” means that the structure is single-ended and does not have the mixed-mode S-parameters. Simulation was done with de-compositional analysis in Simbeor software and measurements are provided by WRT. The models are measurements are used from CMP-28 Simbeor Kit Rev. 4 with all data available at https://drive.google.com/drive/folders/0B6jLiKYCgxAnbFE0WFRmamxvLVE?usp=sharing.

We can see that there is much better similarity at lower frequencies (10 GHz column) and it degrades with larger bandwidths (35 and 50 GHz columns). Note that some structures have lower SPP – MS_SE_GND_Voids_J74_J75 for instance – this is because of the loss of localization. Complete Kit with all data and plots can be downloaded for further comparisons and experiments. See more in [4].

“Sink or swim” approach [1] was validated with EvR-1 platform first introduced in [6] and later used in [2]. The last step of the approach is to simulate every single structure on the validation platform with identified material models and manufacturing adjustments, but without any “calibration”, “tuning” or “tweaking” and observe the correlation. Results of the visual analysis in provided in [2]. Now we can do it with the new SPS metric automatically computed with Simbeor SDK. The results are shown in the table below:

Further analysis of this data is provided in [4]. Complete EvR-1 Kit is available https://drive.google.com/drive/folders/1Rm-QpROluiQ_fsIfpetCt8hu8PtfZfkz?usp=sharing

References:

1.      Y. Shlepnev, Sink or swim at 28 Gbps, The PCB Design Magazine, October 2014, p. 12-23. - #2014_05 at https://www.simberian.com/AppNotes.php

2.      M. Marin, Y. Shlepnev, Systematic approach to PCB interconnects analysis to measurement validation, Proc. of 2018 IEEE Int. Symp. on EMC and SIPI, July 30- August 3, 2018, Long Beach, CA. - #2018_07 at https://www.simberian.com/AppNotes.php

3.      A.P. Duffy, A. Orlandi, G. Zhang, Review of the Feature Selective Validation Method (FSV). Part I – Theory, IEEE Trans. On EMC, vol. 60, No. 4, August 2018, p. 814-821.

4.      Y. Shlepnev, S-Parameters Similarity Metric, Simberian Inc., App note #2021_05, Apr. 24, 2021. - #2021_05 at https://www.simberian.com/AppNotes.php

5.      Guide to CMP-28/32 Simbeor Kit, CMP-28 Rev. 4, Sept. 2014.

6.      M. Marin, Y. Shlepnev, 40 GHz PCB Interconnect Validation: Expectation vs. Reality, DesignCon2018, January 31, 2018, Santa Clara, CA. - #2018_01 at https://www.simberian.com/AppNotes.php