Vector Stochastic Differential Equations Used to Electrical Networks with Random Parameters

Edita Kolářová, Lubomír Brančík


In this paper we present an application of the Itô stochastic calculus to the problem of modelling RLC electrical circuits. The deterministic model of the circuit is replaced by a stochastic model by adding a noise term to various parameters of the circuit. The analytic solutions of the resulting stochastic integral equations are found using the multidimensional Itô formula. For the numerical simulations in the examples we used MATLABr. The SDE approach has its perspectives in the simulation even higher-order circuits representing more complex physical systems, as their real implementations are often subject to a number of random effects. An example for a transmission line lumped-parameter model is provided.

Full Text:



L. Arnold, Stochastic Differential Equations: Theory and Applications, New York: John Wiley & Sons, 1974.

B. Øksendal, Stochastic Differential Equations, An Introduction with Applications, New York: Springer-Verlag, 2000.

P. Kloeden, E. Platen, and H. Schurz, Numerical Solution of SDE Through Computer Experiments, New York: Springer-Verlag, 1997.

S. Cyganowski, P. Kloeden, and J. Ombach, From Elementary Probability to Stochastic Differential Equations with Maple, New York: Springer-Verlag, 2000.

D. Ham, and A. Hajimiri, ”Complete Noise Analysis for CMOS Switching Mixers via Stochastic Differential Equations,” in Proc. IEEE 2000 Custom Integrated Circuits Conference, Orlando, pp. 439-442., 2000.

K. Wang, and M. L. Crow, ”Numerical Simulation of Stochastic Differential Algebraic Equations for Power System Transient Stability with Random Loads,” in Proc. 2011 IEEE Power and Energy Society General Meeting, San Diego, pp. 1-8, 2011.

A. Zjajo, Q. Tang, M. Berkelaar, J. P. de Gyvez, A. Di Bucchianico,and N. van der Meijs, ”Stochastic Analysis of Deep-Submicrometer CMOS Process for Reliable Circuits Designs,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 58, no. 1, pp. 164-175, 2011.

E. Kolářová, ”Modelling RL Electrical Circuits by Stochastic Diferential Equations,” in Proceedings of International Conference on Computer as a Tool, Belgrade (Serbia and Montenegro): IEEE R8, pp. 1236-1238, 2005.

E. Kolářová, ”Statistical estimates of stochastic solutions of RL electrical circuits,” in Proc. of the IEEE International Conference on Industrial Technology ICIT06, Mumbai, pp. 2546-2550, 2006.

N. S. Patil, B. G. Gawalwad, and S. N. Sharma, ”A random inputdriven resistor-capacitor series circuit,” in Proc. 2011 International Conference on Recent Advancements in Electrical, Electronics and Control Engineering, Sivakasi, pp. 100-103, 2011.

F. Rahman, and N. Parisa, ”A stochastic perspective of RL electrical circuit using different noise terms,” COMPEL: The International Journal

for Computation and Mathematics in Electrical and Electronic Engineering, vol. 30, no. 2, pp. 812-822, 2011.

C. K. Cheng, J. Lillis, S. Lin, and N. Chang, Interconnect Analysis and Synthesis, New York: John Wiley & Sons, 2000.

M. P. Li, Jitter, Noise, and Signal Integrity at High-Speed, Upper Saddle River: Prentice Hall, 2007.

E. Kolářová, and L. Brančík, ”Application of stochastic differential equations in second-order electrical circuits analysis,” Przeglad Elektrotechniczny, vol. 88, no. 7a, 2012, pp. 103-107.

L. Brančík, ”Time and Laplace-domain methods for MTL transient and sensitivity analysis,” COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 30, no. 4, pp. 1205-1223, 2011.

L. Brančík, and E. Kolářová, ”Analysis of higher-order electrical circuits with noisy sources via stochastic differential equations approach,” in Proc. of 22nd Int. Conf. Radioelektronika’2012, Brno, 2012, pp. 81-84.

E. Kolářová, and L. Brančík, ”Vector linear stochastic differential equations and their applications to electrical networks,” in Proc. of 35th Int. Conf. on Telecommunications and Signal Processing TSP2012, Prague, 2012, pp. 311-315.

D. V. Ginste, D. De Zutter, D. Deschrijver, T. Dhaene, P. Manfredi, and F. Canavero, ”Stochastic Modeling-Based Variability Analysis of On-Chip Interconnects”, IEEE Trans. on Components, Packaging and Manuf. Technology, vol. 2, no. 7, 2012, pp. 1182-1192.

D. Spina, F. Ferranti, T. Dhaene, L. Knockaert, G. Antonini, and D. V. Ginste, ”Variability analysis of multiport systems via polynomial-chaos expansion”, IEEE Trans. on MTT, vol. 60, no. 8, 2012, pp. 2329-2338.

D. V. Ginste, D. D. Zutter, D. Deschrijver, T. Dhaene, and F. Canavero, ”Macromodeling based variability analysis of an inverted embedded

microstrip line” in Proc. IEEE 20th Int. Conf. Elect. Perform. Electron. Packag. Syst., San Jose, CA, 2011, pp. 153-156.

L. Brančík, and E. Kolářová, ”Stochastic differential equations approach in the analysis of MTLs with randomly varied parameters,” in Proc. of 19th IEEE International Conference on Electronics, Circuits, and Systems ICECS2012, Sevilla, 2012, pp. 725-728



  • There are currently no refbacks.