This paper introduces a tensorial analysis of networks (TAN) applied to a tree asymmetrical structure. To illustrate the TAN concept easily, the present investigation is applied to a three-port structure represented by a Y-tree topology. The unfamiliar method of TAN circuit modelling is elaborated from the graph topology. The fast formulation of the Y-matrix model of the structure is established from branch and mesh space TAN analyses. The TAN model is validated with commercial tool simulation and measurements from DC up to 0.5 GHz in the frequency domain and two different waveform signals in the time domain. The proof of concept circuit is implemented in microstrip technology on an FR4-epoxy dielectric substrate. Mapping sensitivity analysis with respect to the Y-tree RLC-parameters is realized by showing that local variations around initial set of R, L, and C do not equally influence reflection and transmission coefficients over the frequency bandwidth. If a similar impact is observed at the lowest frequency, maximum variations up to 250% show the importance of parameters ranking to improve both microstrip design and modelling.
2. Ney, M., "Method of moments as applied to electromagnetic problems," IEEE Trans. MTT, Vol. 33, No. 10, 972-980, Oct. 1985.
3. Jin, J., The Finite Element Method in Electromagnetics, John Wiley & Sons, New York, USA, 1993.
4. Rizzoli, V., A. Costanzo, F. Mastri, and A. Neri, "A general SPICE model for arbitrary linear dispersive multiport components described by frequency-domain data," Proc. 2003 IEEE MTT-S Int. Microwave Symp. Digest, Vol. 1, 9-12, Philadelphia, PA, USA, Jun. 8-13, 2003.
5. Krishna, K. S. R., J. L. Narayana, and L. P. Reddy, "ANN models for microstrip line synthesis and analysis," Int. J. Elect. Syst. Sci. Eng., Vol. 1, 196-200, 2008.
6., , https://www.3ds.com/products-services/simulia/products/cst-studio-suite/, accessed 2019.
7., , https://www.ansys.com/products/electronics/ansys-hfss, accessed 2019.
8., , https://altairhyperworks.com/product/FEKO/Applications-Antenna-Design, accessed 2019.
9., , https://www.keysight.com/us/en/assets/7018-02343/brochures/5990-4819.pdf, accessed 2019.
10., , EM/EMC Simulation Software, https://www.emcos.com/?product-types=em-simulation-software, accessed 2019.
11. Schuster, C. and W. Fichtner, "Parasitic modes on printed circuit boards and their effects on EMC and signal integrity," IEEE Trans. EMC, Vol. 43, No. 4, 416-425, Nov. 2001.
12. Archambeault, R., C. Brench, and S. Connor, "Review of printed-circuit-board level EMI/EMC issues and tools," IEEE Trans. EMC, Vol. 52, No. 2, 455-461, May 2010.
13. Kim, J. and E. Li, "Special issue on PCB level signal integrity, power integrity, and EMC," IEEE Trans. EMC, Vol. 52, No. 2, 246-247, May 2010.
14. Ruehli, A. E. and A. C. Cangellaris, "Progress in the methodologies for the electrical modeling of interconnects and electronic packages," Proceedings of the IEEE, Vol. 89, No. 5, 740-771, 2001.
15. Ruan, A., J. Yang, L. Wan, B. Jie, and Z. Tian, "Insight into a generic interconnect resource model for Xilinx Virtex and Spartan series FPGAs," IEEE Trans. CAS-II: Express Briefs, Vol. 60, No. 11, 801-805, Nov. 2013.
16. Buckwalter, J. F., "Predicting microwave digital signal integrity," IEEE Trans. Advanced Packaging, Vol. 32, No. 2, 280-289, May 2009.
17. Jun, F., X. Ye, J. Kim, B. Archambeault, and A. Orlandi, "Signal integrity design for high-speed digital circuits: Progress and directions," IEEE Trans. EMC, Vol. 52, No. 2, 392-400, May 2010.
18. Ruehli, A. E. and A. C. Cangellaris, "Progress in the methodologies for the electrical modeling of interconnects and electronic packages," Proc. of the IEEE, Vol. 89, No. 5, 740-771, 2001.
19. Charlet, F. and J. F. Carpentier, "Extraction of 3D interconnect impedances using edge elements without gauge condition," Proc. Int. Conf. on Simulation of Semiconductor Processes and Device, 143-146, Kobe, Japan, Sep. 4-6, 2002.
20. Chen, M., D. Shi, Y. Li, L. Zhu, and H. Liu, "Research on branches group based method for adding mutual inductance branches to Y -matrix and Z-matrix," Proc. 2014 IEEE PES General Meeting Conference & Exposition, 1-5, National Harbor, MD, USA, Jul. 27-31, 2014.
21. Wojnowski, M., M. Engl, and R. Weigel, "Considerations on impedance matrix determination for accurate passive device characterization," Proc. 2007 IEEE Workshop SPI, 117-120, Genova, Italy, May 13-16, 2007.
22. Tu, C., J. Bao, Y. Du, and W. Wu, "An improved design method for asymmetric RF MEMS tunable filter utilizing admittance matrix," Proc. 2010 IEEE Int. Conf. on Microwave and Millimeter Wave Technology, 1766-1769, Chengdu, China, May 8-11, 2010.
23. Chiariello, A. G., A. Girardi, C. Iorio, R. Izzi, T. Lessio, A. Maffucci, and S. Ventre, "Efficient evaluation of the frequency-dependent impedance matrix of full-package structures," Proc. 2010 IEEE 14th Workshop on SPI, 127-130, Hildesheim, Germany, May 9-12, 2010.
24. Ymeri, H., B. Nauwelaers, K. Maex, D. De Roest, S. Vandenberghe, and M. Stucchi, "Admittance matrix calculations of on-chip interconnects on lossy silicon substrate using multilayer Green’s function," Proc. 2001 IEEE Topical Meeting on Silicon Monolithic Integrated Circuits in RF Systems. Digest of Papers (IEEE Cat. No. 01EX49 , 50-59, Ann Arbor, MI, USA, Sep. 14, 2001.
25. Sun, Y.-Y., "Immittance matrices of multiconductor transmission lines," Journal of the Franklin Institute, Vol. 307, No. 1, 59-67, Jan. 1979.
26. Xu, J. and Y.-H. Lv, "System-level construction of multiconductor transmission line inductance matrix," Proc. 2009 3rd IEEE Int. Symp. on Microwave, Antenna, Propagation and EMC Technologies for Wireless Communications, 903-906, Beijing, China, Oct. 27-29, 2009.
27. Hou, R. and C. Taibin, "Y parameter matrix and calculation method built for multi-port network based on graph theory," Proc. 2011 IEEE 3rd International Conference on Communication Software and Networks, 621-623, Xi’an, China, May 27-29, 2011.
28. Ravelo, B., "Behavioral model of symmetrical multi-level T-tree interconnects," Progress In Electromagnetics Research B, Vol. 41, 23-50, 2012.
29. Kron, G., Tensor Analysis of Networks, Wiley, New York; Chapman & Hall, London, 1939.
30. Maurice, O., A. Reineix, P. Hoffmann, B. Pecqueux, and P. Pouliguen, "A formalism to compute the electromagnetic compatibility of complex networks," Advances in Applied Science Research, Vol. 2, No. 5, 439-448, 2011.
31. Maurice, O., Elements of Theory for Electromagnetic Compatibility and Systems, Bookelis, Aix en Provence, France, 2017.
32. Ravelo, B. and O. Maurice, "Kron-Branin modeling of Y-Y-tree interconnects for the PCB signal integrity analysis," IEEE Trans. on Electromagnetic Compatibility, Vol. 59, No. 2, 411-419, Apr. 2017.
33. Cholachue, C., B. Ravelo, A. Simoens, and A. Fathallah, "Fast S-parameter TAN model of N-port lumped structures," IEEE Access, Vol. 7, No. 1, 72505-72517, Dec. 2019.
34. Xu, Z., Y. Liu, B. Ravelo, and O. Maurice, "Modified Kron’s TAN modeling of 3D multilayer PCB," Proc. of 11th International Workshop on Electromagnetic Compatibility of Integrated Circuits, EMC Compo 2017, 242-247, St. Petersburg, Russia, Jul. 4-8, 2017.
35. Xu, Z., J. Fan, and O. Maurice, "Sensitivity analysis of PCB interconnect and package with TAN formalism," Proc. of 2019 12th International Workshop on the Electromagnetic Compatibility of Integrated Circuits (EMC Compo), 81-83, Hangzhou, China, 2019.
36. Xu, Z., Y. Liu, B. Ravelo, and O. Maurice, "Multilayer power delivery network modeling with modified Kron’s method (MKM)," Proc. of 16th Int. Symposium on Electromagnetic Compatibility (EMC) Europe 2017, 1-6, Angers, France, Sep. 4-8, 2017.
37. Xu, Z., Y. Liu, B. Ravelo, J. Gantet, N. Marier, and O. Maurice, "Direct time-domain TAN model of 3D multilayer hybrid PCB: Experimental validation," IEEE Access, Vol. 6, No. 1, 60645-60654, Dec. 2018.
38. Gupta, K. C., R. Garg, and I. Bahl, Microstrip Lines and Slotlines, Artech, Dedham, MA, 1979.
39. Hammerstad, E. and O. Jensen, "Accurate models for microstrip computer aided design," Proc. 1980 IEEE MTT-S Int. Microwave Symp. Digest, 407-409, Washington, DC, USA, May 28-30 1980.
40. Frickey, D. A., "onversions between S, Z, Y, h, ABCD, and T parameters which are valid for complex source and load impedances," IEEE Trans. on MTT, Vol. 42, No. 2, 205-211, Feb. 1994.
41. Saltelli, A., S. Tarantola, F. Campolongo, and M. Ratto, Sensitivity Analysis in Practice: A Guide to Assessing Scientific Models, John Wiley & Sons, 2004.