Abstract
The work concerns the application of a solver based on SubIval (the subinterval-based method for computations of the fractional derivative in initial value problems). A general form of a system of equations is considered. The system is one that can be formulated for a circuit with fractional elements, nonlinear elements and elements that are both fractional and nonlinear. A few details are given on the computations that need to be performed in each time step. An example with fractional, nonlinear elements is introduced to display the usefulness of the solver. The results obtained through the solver are compared with ones obtained through a harmonic balance methodology (steady state solution). Finally, a measure of the accuracy of the results is introduced and computed for the selected example.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Oprzȩdkiewicz, K.: Accuracy estimation of digital fractional order PID controller. In: Theory and Applications of Non-integer Order Systems, pp. 265–275 (2017)
Zagórowska, M.: Analysis of performance indicators for tuning non-integer order controllers. In: Theory and Applications of Non-Integer Order Systems, pp. 307–317 (2017)
Spałek, D.: Synchronous generator model with fractional order voltage regulator PI\(^{b}\)D\(^{a}\). Acta Energetica 2(23), 78–84 (2015)
Mercorelli, P.: A discrete-time fractional order PI controller for a three phase synchronous motor using an optimal loop shaping approach. In: Theory and Applications of Non-Integer Order Systems, pp. 477–487 (2017)
Bauer, W., Kawala-Janik, A.: Implementation of bi-fractional filtering on the arduino uno hardware platform. In: Theory and Applications of Non-Integer Order Systems, pp. 419–428 (2017)
Kawala-Janik, A., Podpora, M., Gardecki, A., Czuczwara, W., Baranowski, J., Bauer, W.: Game controller based on biomedical signals. In: 2015 20th International Conference on Methods and Models in Automation and Robotics (MMAR), pp. 934–939 (2015)
Garrappa, R., Maione, G.: Fractional prabhakar derivative and applications in anomalous dielectrics: a numerical approach. In: Theory and Applications of Non-Integer Order Systems, pp. 429–439 (2017)
Mescia, L., Bia, P., Caratelli, D.: Fractional derivative based FDTD modeling of transient wave propagation in Havriliak-Negami media. IEEE Trans. Microw. Theory Tech. 62(9), 1920–1929 (2014)
Litak, G., Ducharne, B., Sebald, G., Guyomar, D.: Dynamics of magnetic field penetration into soft ferromagnets. J. Appl. Phys. 117(24), 243907 (2015)
Brociek, R., Słota, D., Wituła, R.: Reconstruction robin boundary condition in the heat conduction inverse problem of fractional order. In: Theory and Applications of Non-Integer Order Systems, pp. 147–156 (2017)
Sierociuk, D., Skovranek, T., Macias, M., Podlubny, I., Petras, I., Dzielinski, A., Ziubinski, P.: Diffusion process modeling by using fractional-order models. Appl. Math. Comput. 257, 2–11 (2015)
Žecová, M., Terpák, J.: Heat conduction modeling by using fractional-order derivatives. Appl. Math. Comput. 257, 365–373 (2015)
Fouda, M.E., Elwakil, A.S., Radwan, A.G., Allagui, A.: Power and energy analysis of fractional-order electrical energy storage devices. Energy 111, 785–792 (2016)
Schäfer, I., Krüger, K.: Modelling of lossy coils using fractional derivatives. Phys. D Appl. Phys. 41, 1–8 (2008)
King, A., Agerkvist, F.T.: State-space modeling of loudspeakers using fractional derivatives. Audio Engineering Society Convention 139. Audio Engineering Society (2015)
Gómez Aguilar, J.F., Hernández, M.M.: Space-time fractional diffusion-advection equation with caputo derivative. Abstr. Appl. Anal. 2014, 8 p. (2014)
Katugampola, U.N.: A new approach to generalized fractional derivatives. Bull. Math. Anal. Appl. 6(4), 1–15 (2014)
Caputo, M.: Linear models of dissipation whose Q is almost frequency independent - II. Geophys. J. Int. 13(5), 529–539 (1967)
Munkhammar, J.D.: Riemann-Liouville fractional derivatives and the Taylor-Riemann series. UUDM Project Rep. 7, 1–18 (2004)
Abdeljawad, T.: On Riemann and Caputo fractional differences. Comput. Math. Appl. 62, 1602–1611 (2011)
Ruszewski, A.: Stability analysis for the new model of fractional discrete-time linear state-space systems. In: Theory and Applications of Non-Integer Order Systems, pp. 381–389 (2017)
Klamka, J., Czornik, A., Niezabitowski, M., Babiarz, A.: Controllability and minimum energy control of linear fractional discrete-time infinite-dimensional systems. In: 11th IEEE International Conference Control & Automation (ICCA), pp. 1210–1214 (2014)
Kaczorek, T., Rogowski, K.: Fractional Linear Systems and Electrical Circuits. Springer, New York (2014)
Mitkowski, W., Skruch, P.: Fractional-order models of the supercapacitors in the form of RC ladder networks. Bull. Pol. Acad. Tech. 61(3), 580–587 (2013)
Kaczorek, T.: Stability analysis for the new model of fractional discrete-time linear state-space systems. In: Theory and Applications of Non-Integer Order Systems, pp. 45–55 (2017)
Momani, S., Noor, M.A.: Numerical methods for fourth order fractional integro-differential equations. Appl. Math. Comput. 182, 754–760 (2006)
Pirkhedri, A., Javadi, H.H.S.: Solving the time-fractional diffusion equation via Sinc–Haar collocation method. Appl. Math. Comput. 257, 317–326 (2015)
Luo, W.-H., Huang, T.-Z., Wu, G.-C., Gu, X.-M.: Quadratic spline collocation method for the time fractional subdiffusion equation. Appl. Math. Comput. 276, 252–265 (2016)
Lubich, C.: Fractional linear multistep methods for Abel-Volterra integral equations of the second kind. Math. Comput. 45, 463–469 (1985)
Cui, M.: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228, 7792–7804 (2009)
Sowa, M.: A subinterval-based method for circuits with fractional order elements. Bull. Pol. Acad. Tech. 62(3), 449–454 (2014)
Sowa, M.: Application of SubIval in solving initial value problems with fractional derivatives. Appl. Math. Comput. 319, 86–103 (2017). (in press)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
A Formulation of State Equations
A Formulation of State Equations
The vector of state variables takes the form:
with the derivative orders:
The vector of additional variables is given by:
The vector of source time functions is:
The current balances contribute to the matrices marked by \(\varvec{M}_\mathrm {I}\), \(\varvec{M}_\mathrm {II}\) and \(\varvec{T}\). For the node with the potential \(V_1\):
for the \(V_2\) node:
while for the node marked by \(V_3\):
and for the \(V_4\) node:
The relations between the marked voltages and the node potentials contribute to \(\varvec{M}_\mathrm {I}\) and \(\varvec{M}_\mathrm {II}\). From the relation between \(V_1\) and \(u_q\):
between \(V_2\) and \(u_\mathrm {NL}\)
between \(V_2\) and \(u_\psi \)
while for \(u_L\) and its terminal potentials:
and for the relation between \(u_\kappa \) and \(V_3\):
The nonlinear dependency \(i_\mathrm {NL}(u_\mathrm {NL})\) describing the response of the nonlinear resistor \(R_\mathrm {NL}\) is given in \(\varvec{F}_\mathrm {NL}(\varvec{w}(t))\) by:
hence the index of this nonlinear function’s argument:
The left-hand side of this equation introduces:
In the same manner the nonlinear function of \(C_q\) introduces:
with:
and:
For the nonlinear function of \(L_\psi \) one obtains:
with:
and:
The differential equation describing the fractional capacitor \(C_\kappa \) introduces:
for the coil L:
while for the fractional, nonlinear capacitor \(C_q\):
and for the fractional, nonlinear coil \(L_\psi \):
Rights and permissions
Copyright information
© 2019 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Sowa, M. (2019). Solutions of Circuits with Fractional, Nonlinear Elements by Means of a SubIval Solver. In: Ostalczyk, P., Sankowski, D., Nowakowski, J. (eds) Non-Integer Order Calculus and its Applications. RRNR 2017. Lecture Notes in Electrical Engineering, vol 496. Springer, Cham. https://doi.org/10.1007/978-3-319-78458-8_19
Download citation
DOI: https://doi.org/10.1007/978-3-319-78458-8_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-78457-1
Online ISBN: 978-3-319-78458-8
eBook Packages: EngineeringEngineering (R0)