# Single Step 2-Port Device De-Embedding Algorithm for Fixture-DUT-Fixture Network Assembly

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}) where N is the order of the matrix) due to the iterative process for the evaluation of the inverse [11]. Once implemented, the solution is applied directly on the raw measured S-parameters of the fixture-DUT-fixture system knowing those of each fixture without any conversion to T-parameters. The analytical solution is an extension of the algorithm developed for the cascading of two S-parameter blocks in [12]. The algorithm is expanded to the case of three blocks (or networks) representing the S-parameters of the left fixture (FL), of the DUT and of the right fixture (FR) based on the cascading of the S-parameter blocks network considered in [12]. The classical method for the cascading of two networks is here expanded to the case of three networks that are identified in the following as the left fixture (FL), the DUT, and the right fixture (FR). The direct solution of the three blocks cascading process gives rise, through few matrix-algebraic manipulations, to a set of four non-linear equations in four auxiliary unknowns that are directly and analytically related to the sought DUT S-parameters (S

_{DUT1,1}, S

_{DUT1,2}, S

_{DUT2,1}, S

_{DUT2,2}). This system of non-linear equations has a closed form solution clear of any numerical error.

_{DUT1,1}, S

_{DUT1,2}, S

_{DUT2,1}, S

_{DUT2,2}) plus their combination coming out from the matrix determinant built along the cascading process. By means of a matrix-algebraic manipulation the problem is reduced to a set of four auxiliary equations in four unknowns and their solution used to find the valid and unique closed form solution of the DUT S-parameters.

## 2. S-Parameter Cascade of Three Networks

#### 2.1. Review of Two-Network S-Parameter Cascading Algorithm

_{α,β}and b

_{α,β}are the incident and reflected power waves respectively [10]. The subscript α = FL, FR denotes if the wave belongs to the FL or FR network; the subscript β = i,o identifies if the wave is at the input or output port of the network.

_{TOT}in (6) from the external end ports FL

_{i}and FR

_{o}of the cascaded FL and FR networks.

#### 2.2. Single Step S-Parameter Cascading Algorithm for Three 2-Port Networks

_{i}and FR

_{o}in Figure 2.

## 3. Single-Step Algorithm for Device under Test (DUT) De-Embedding

## 4. Validation of the Proposed Single-Step De-Embedding

_{TOT}from Port 1 to Port 2 in Figure 3a have been measured in [15] up to 50 GHz. All the four blocks of S-parameters (for the original DUT, for the two fixtures and for the entire system from Port 1 to Port 2) in this work have been considered as input data for this work.

_{2}from (20)) from the knowledge of S

_{TOT}(from Port 1 to Port 2 in Figure 3a) and comparing them with other independent solutions. The de-embedded DUT S-parameters from (20) (named “Single-Step De-Embed.” in Figure 4) are compared:

- to those obtained by applying the classic standard two-step S-to-T and T-to-S parameter conversion named “S-T conv. De-embedding” in Figure 4;

_{21}, S

_{11}, and S

_{22}are reported in Figure 4.

_{ij}by each method and the same S-parameter of the standalone original DUT.

_{ij}by each method and the same S-parameter of the standalone original DUT.

_{22}parameter whose error, although small, is not acceptable, since it is only one order of magnitude smaller than its absolute value. Both tables also intend to show that, from the accuracy point of view, the single step method is as accurate as the standard one performed in multiple steps. The main advantages of the new algorithm can be summarized as: (1) it uses a closed form solution without any approximation or inherent numerical error; (2) it is performed in one step so it is about 3× faster than the classic one: given the normalized execution time of the classic standard de-embedding procedure is 1, the normalized execution time of the single step procedure is 0.333 on the same hardware platform. This latter feature is very important when the de-embedding procedure should be repeated several times, as there are many frequencies of the spectrum of the sought DUT scattering parameters (usually several thousands).

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Stenarson, J.; Yhland, K. An overview of VNA calibration and uncertainty evaluation techniques. Ger. Microw. Conf.
**2008**, 1, 1–4. [Google Scholar] - Stumper, U. Uncertainties of VNA S-Parameter Measurements Applying the TAN Self-Calibration Method. IEEE Trans. Instrum. Meas.
**2007**, 56, 597–600. [Google Scholar] [CrossRef] - Liu, Y.; Yong, S.; Gao, H.; Hinaga, S.; Padilla, D.; Yanagawa, D.; Drewniak, J.L.; Khilkevich, V. S-Parameter De-Embedding Error Estimation Based on the Statistical Circuit Models of Fixtures. IEEE Trans. Electromagn. Compat.
**2020**, 62, 1459–1467. [Google Scholar] [CrossRef] - De Paulis, F.; Piersanti, S.; Wang, Q.; Cho, J.; Erickson, N.; Achkir, B.; Fan, J.; Drewniak, J.; Orlandi, A. TEM-Like Launch Geometries and Simplified De-embedding for Accurate Through Silicon Via Characterization. IEEE Trans. Instrum. Meas.
**2017**, 66, 792–801. [Google Scholar] [CrossRef] - Resso, M.; Bogatin, E.; Vatsyayan, A. A new method to verify the accuracy of de-embedding algorithms. In Proceedings of the 2016 IEEE MTT-S Latin America Microwave Conference (LAMC), Puerto Vallarta, Mexico, 12–14 December 2016. [Google Scholar]
- Salnikov, A.S.; Dobush, I.M.; Bilevich, D.V.; Popov, A.A.; Kalentyev, A.A.; Goryainov, A.E. A Study of Connectors and Feed Lines De-Embedding Techniques for PCB Microwave Components S-Parameters Measurements Up To 50 Ghz. In Proceedings of the 2020 Dynamics of Systems, Mechanisms and Machines (Dynamics), Omsk, Russia, 10–12 November 2020. [Google Scholar]
- Chen, B.; Ye, X.; Samaras, B.; Fan, J. A novel de-embedding method suitable for transmission-line measurement. In Proceedings of the 2015 Asia-Pacific Symposium on Electromagnetic Compatibility (APEMC), Taipei, Taiwan, 25–29 May2015. [Google Scholar]
- Barnes, H.; Bogatin, E.; Moreira, J. Development of a PCB kit for s-parameter de-embedding algorithms verification. In Proceedings of the 2017 IEEE International Symposium on Electromagnetic Compatibility & Signal/Power Integrity (EMCSI), Washington, DC, USA, 7–11 August 2017. [Google Scholar]
- Cho, H.; Burk, D. A three-step method for the de-embedding of high-frequency S-parameter measurements. IEEE Trans. Electron Devices
**1991**, 38, 1371–1375. [Google Scholar] [CrossRef] - Mavaddat, R. Network Scattering Parameters; World Scientific: Jefferson, NJ, USA, 1996. [Google Scholar]
- Sadiku, M.N.O. Numerical Techniques in Electromagnetics, 2nd ed.; CRC: New York, NY, USA, 2000. [Google Scholar]
- De Paulis, F.; Zhang, Y.-J.; Fan, J. Signal/Power Integrity Analysis for Multilayer Printed Circuit Boards Using Cascaded S-Parameters. IEEE Trans. Electromagn. Compat.
**2010**, 52, 1008–1018. [Google Scholar] [CrossRef] - De Paulis, F.; Wang-Lee, T.; Mellitz, R.; Resso, M.; Rabinovich, R.; Danzy, O.J. Backplane channel design exploration at 112 Gbps using channel operating margin (COM). In Proceedings of the 2020 IEEE International Symposium on Electromagnetic Compatibility & Signal/Power Integrity (EMCSI), Reno, NJ, USA, 28 July–28 August 2020. [Google Scholar]
- SAMTEC. Available online: https://www.samtec.com/connectors/high-speed-board-to-board/high-density-arrays/novaray (accessed on 18 May 2021).
- De Paulis, F.; Wang-Lee, T.; Resso, M.; Mellitz, R.; Rabinovich, R.; Danzy, O. Validation and Performance Evaluation of High Speed Connector Model for Channel Design at 56 Gbps and Above. In Proceedings of the 2020 IEEE 24th Workshop on Signal and Power Integrity (SPI), Cologne, Germany, 17–20 May 2020. [Google Scholar]
- Mathworks. Matlab User Manual. Available online: https://it.mathworks.com/help/optim/ug/fsolve.html (accessed on 1 May 2021).

**Figure 1.**Cascade of two 2-port S-parameter networks: “FL” stands for left feature network and “FR” stands for right feature network.

**Figure 3.**(

**a**) Overview of the full de-embedding setup, (

**b**) details of the FL and FR fixtures, (

**c**) complete fixtures from the original multiport model.

**Figure 4.**Comparison of the S-parameters obtained by: 1. the standalone original device under test (DUT) considered as reference (continuous line “Orig. DUT”), 2. the classic standard two-steps S-to-T and T-to-S parameter conversion (dashed line “S-T conv. De-embed”), 3. the single-step procedure (circles “Single-Step De-embed.”) and 4.by the numerical iterative solution (dash-dot line “Numerical calc.”). (

**a**) S

_{21}, (

**b**) S

_{11}, (

**c**) S

_{22}.

Method | S_{11} | S_{22} | S_{21} |
---|---|---|---|

Classic standard three-step (S-T conv. De-embedding) | 6.15 × 10^{−9} + j × 1.08 × 10^{−8} | 1.09 × 10^{−9} + j × 3.9 × 10^{−9} | 2.04 × 10^{−8} + j × 3.8 × 10^{−9} |

Proposed single step (Single-Step De-embed.) | 6.15 × 10^{−9} + j × 1.08 × 10^{−8} | 1.09 × 10^{−9} + j × 3.9 × 10^{−9} | 2.04 × 10^{−8} + j × 3.8 × 10^{−9} |

Numerical solution (Numerical calc.) | 1.08 × 10^{−5} + j × 1.4 × 10^{−5} | 0.03 + j × 0.025 | 1.30 × 10^{−4} + j × 7.33 × 10^{−4} |

Method | S_{11} | S_{22} | S_{21} |
---|---|---|---|

Classic standard three-step (S-T conv. De-embedding) | 9.26 × 10^{−18} | 7.76 × 10^{−18} | 2.09 × 10^{−17} |

Proposed single step (Single-Step De-embed.) | 9.26 × 10^{−18} | 7.76 × 10^{−18} | 2.09 × 10^{−17} |

Numerical solution (Numerical calc.) | 3.29 × 10^{−11} | 2.20 × 10^{−4} | 6.51 × 10^{−8} |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Scafati, S.; Pellegrino, E.; de Paulis, F.; Olivieri, C.; Drewniak, J.; Orlandi, A.
Single Step 2-Port Device De-Embedding Algorithm for Fixture-DUT-Fixture Network Assembly. *Electronics* **2021**, *10*, 1275.
https://doi.org/10.3390/electronics10111275

**AMA Style**

Scafati S, Pellegrino E, de Paulis F, Olivieri C, Drewniak J, Orlandi A.
Single Step 2-Port Device De-Embedding Algorithm for Fixture-DUT-Fixture Network Assembly. *Electronics*. 2021; 10(11):1275.
https://doi.org/10.3390/electronics10111275

**Chicago/Turabian Style**

Scafati, Simone, Enza Pellegrino, Francesco de Paulis, Carlo Olivieri, James Drewniak, and Antonio Orlandi.
2021. "Single Step 2-Port Device De-Embedding Algorithm for Fixture-DUT-Fixture Network Assembly" *Electronics* 10, no. 11: 1275.
https://doi.org/10.3390/electronics10111275