Fractional RC and LC Electrical Circuits

Abstract

In this paper we propose a fractional differential equation for the electrical RC and LC circuit in terms of the fractional time derivatives of the Caputo type. The order of the derivative being considered is 0 <  ≤ 1. To keep the dimensionality of the physical parameters R, L, C the new parameter σ is introduced. This parameter characterizes the existence of fractional structures in the system. A relation between the fractional order time derivative  and the new parameter σ is found. The numeric Laplace transform method was used for the simulation of the equations results. The results show that the fractional differential equations generalize the behavior of the charge, voltage and current depending of the values of . The classical cases are recovered by taking the limit when  = 1. An analysis in the frequency domain of an RC circuit shows the application and use of fractional order differential equations.

Time Dimensional Consistency Aware Analysis of Voltage Mode and Current Mode Active Fractional Circuits

“…By carefully considering (23), v o (t) due to DC source voltage can be given in an alternative manner as follows According to [25], T αv stands for the fractional time constant of the candidate voltage mode active fractional circuit. It determines the dynamic of such a circuit as it is defined as the time instant where v o (t) due to DC source with V > v o (0)/(1 + (R 2 /R 1 )) rises to approximately 63.2% of its final value.…”

“…By including such a parameter in the fractional derivative term, the time dimension of the fractional derivative term with this new parameter becomes sec −1 . This motivates electrical engineers to apply the fractional time component parameter included fractional derivative in the analyses of passive fractional circuits [25] [26] which are electrical systems, for obtaining the time dimensional consistency to the conventional derivative. However, similar analysis of active fractional circuits has never been performed, despite the previous attempts to analyse the active fractional circuits with FDE [22], [23].…”

“…By applying different source terms, including the AC sources, to the obtained solutions, the response of both voltage and current mode circuits have been determined and the behaviours of the circuits have been analysed. Moreover, we determine the fractional time constants [25] and the pole locations in the F-plane of these circuits. We also analyse their dynamic behaviours and stabilities.…”

Inclusion of frequency‐dependent parameters in power transmission lines simulation using harmonic analysis and proper generalized decomposition

“…As explained earlier, time integration of such problems is almost impossible to achieve, and hence, PGD provides real accurate option to solve such problems. The problem is similar to the case of fractional derivatives, which is similar to the fractional RLC circuit …”

“…The problem is similar to the case of fractional derivatives, which is similar to the fractional RLC circuit. 25 Equation 27 with R and L as functions of frequency, where R( ) and L( ) are given by the expressions of Equations 18 and 19, reads as 2V (x, )…”