Dynamic Characterization of Typical Electrical Circuits via Structural Properties

Authors

Sofía Avila-Becerril, Gerardo Espinosa-Pérez, Paul Fernandez

Abstract

The characterization of a class of electrical circuits is carried out in terms of both stability properties and steady-state behavior. The main contribution is the interpretation of the electrical topology (how the elements that conform the circuits are interconnected) in terms of mathematical properties derived from the structure of their models. In this sense, at what extent the topology by itself defines the dynamic behavior of the systems is explained. The study is based on the graph theory allowing capturing, departing from the well-known Kirchhoff laws, the topology of the circuits into several matrices with specific structure. The algebraic analysis of these matrices permits identifying conditions that determine whether the system is stable in the sense of Lyapunov and the kind of steady-state behavior that it exhibits. The approach is mainly focused on typical topologies widely used in practice, namely, radial, ring, and mesh networks.

Citation

• Journal: Mathematical Problems in Engineering
• Year: 2016
• Volume: 2016
• Issue:
• Pages: 1–13
• Publisher: Wiley
• DOI: 10.1155/2016/7870462

BibTeX

@article{Avila_Becerril_2016,
  title={{Dynamic Characterization of Typical Electrical Circuits via Structural Properties}},
  volume={2016},
  ISSN={1563-5147},
  DOI={10.1155/2016/7870462},
  journal={Mathematical Problems in Engineering},
  publisher={Wiley},
  author={Avila-Becerril, Sofía and Espinosa-Pérez, Gerardo and Fernandez, Paul},
  year={2016},
  pages={1--13}
}

Download the bib file

References

• Brayton, R. K. & Moser, J. K. A theory of nonlinear networks. I. Quarterly of Applied Mathematics vol. 22 1–33 (1964) – 10.1090/qam/169746
• (1969)
• Weiss, L. & Mathis, W. A Hamiltonian formulation for complete nonlinear RLC-networks. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications vol. 44 843–846 (1997) – 10.1109/81.622990
• Maschke, B. M., van der Schaft, A. J. & Breedveld, P. C. An intrinsic Hamiltonian formulation of the dynamics of LC-circuits. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications vol. 42 73–82 (1995) – 10.1109/81.372847
• Jeltsema, D. & Scherpen, J. M. A. A dual relation between port-Hamiltonian systems and the Brayton–Moser equations for nonlinear switched RLC circuits. Automatica vol. 39 969–979 (2003) – 10.1016/s0005-1098(03)00070-0
• van der Schaft, A. J. & Maschke, B. M. Port-Hamiltonian Systems on Graphs. SIAM Journal on Control and Optimization vol. 51 906–937 (2013) – 10.1137/110840091
• Stykel, T. Balancing-Related Model Reduction of Circuit Equations Using Topological Structure. Lecture Notes in Electrical Engineering 53–83 (2011) doi:10.1007/978-94-007-0089-5_3 – 10.1007/978-94-007-0089-5_3
• Ortega, R., Jeltsema, D. & Scherpen, J. M. A. Power shaping: A new paradigm for stabilization of nonlinear RLC circuits. IEEE Transactions on Automatic Control vol. 48 1762–1767 (2003) – 10.1109/tac.2003.817918
• Fiaz, S., Zonetti, D., Ortega, R., Scherpen, J. M. A. & van der Schaft, A. J. A port-Hamiltonian approach to power network modeling and analysis. European Journal of Control vol. 19 477–485 (2013) – 10.1016/j.ejcon.2013.09.002
• (1994)
• van der Schaft, A. Characterization and partial synthesis of the behavior of resistive circuits at their terminals. Systems & Control Letters vol. 59 423–428 (2010) – 10.1016/j.sysconle.2010.05.005
• Ortega, R., van der Schaft, A., Castanos, F. & Astolfi, A. Control by Interconnection and Standard Passivity-Based Control of Port-Hamiltonian Systems. IEEE Transactions on Automatic Control vol. 53 2527–2542 (2008) – 10.1109/tac.2008.2006930
• (1976)
• Bollobás, B. Modern Graph Theory. Graduate Texts in Mathematics (Springer New York, 1998). doi:10.1007/978-1-4612-0619-4 – 10.1007/978-1-4612-0619-4
• (1979)
• (2011)
• (1999)
• Jayawardhana, B., Ortega, R., García-Canseco, E. & Castaños, F. Passivity of nonlinear incremental systems: Application to PI stabilization of nonlinear RLC circuits. Systems & Control Letters vol. 56 618–622 (2007) – 10.1016/j.sysconle.2007.03.011
• (1965)
• Eminoglu∗, U. & Hocaoglu∗∗, M. H. A NETWORK TOPOLOGY-BASED VOLTAGE STABILITY INDEX FOR RADIAL DISTRIBUTION NETWORKS. International Journal of Power and Energy Systems vol. 29 (2009) – 10.2316/journal.203.2009.2.203-4280