Modeling two loops RLC circuit AC power source using symbolic arithmetic differential equations

ABSTRACT


INTRODUCTION
Due to experimental or numerical analyses on the behavior between a dissipative and conservative system and the absence of fractionalized systematic techniques, recent research has been diverted from the importance of the recent fractional derivatives including the non-singular kernel with non-locality and the singular kernel with the locality.The mathematical representation of an ordinary differential equation (DE) with second-order cubic nonlinearity is the Van-der-Pol equation.The Van-der-Pol equation has been given a time delay in several investigations.Theresistor-inductor-capacitor (RLC) circuit differential equation is derived as a delay differential equation in this study together with the Van der Pol model differential equation [1].Analytical solutions for the Caputo-Fabrizio, Liouville-Caputo, and new Mittag-Leffler function-based fractional derivative to describe the electrical RLC circuit model were previously discussed.The fractional differential equations take different sources into account.When the fractional order equals 1, the conventional behaviors are restored [2].Dynamical system approaches and Melnikov theory can be used to examine tiny amplitude perturbations of some implicit differential equations appearing in RLC circuits [3].A steady-state process in an RLC circuit with power sources operating at unrelated frequencies is also considered [4].An expansion of an ordinary differential equation is taken into consideration in order to achieve the periodic steady-state behavior.This expansion is based on changing from ordinary differential equations to partial differential equations with two-time variables by adding an additional time variable.The two-dimensional Laplace transform is used to solve the obtained differential equations.Active power and frequency responses for the domain of two time variables are specified by the use of double integral formulas for a transfer function.The voltage and current amplitude-frequency properties of the RLC circuit can be given in the domain of two variables.A nonlinear fractional derivative based Volterra integral-differential equation with Caputo, several kernels, and numerous constant delays is also considered to look into the qualitative properties of solutions to this equation, including the boundedness of nonzero solutions and the Mittag-Leffler stability, uniform stability, and asymptotic stability of the zero solution [5].The Lyapunov-Razumikhin approach and selecting an acceptable Lyapunov function are the methods employed in the proofs of these theorems related to an RLC circuit.
A modified Laplace transform approach to find solutions to a series-connected simple electric circuits (RLC) model of linear differential equations (DEs) was presented in [6].Although the study suggested that non-homogenous second order linear differential equations in the form of electric charge equations can be solved over time using a modified Laplace transform method, only homogeneous second order linear DEs was considered.Certain approximation techniques for generated nonlinear terms of characteristic exponents to offer the stability analysis of delay integral-differential equations with fractional order derivatives were discussed in [7].Such methods allow for the proof of the presence of some analytical solutions close to their equilibrium points.The formulas for the critical time delay and critical frequency are determined through the construction of stability charts to describe general RLC circuits that expose the delay and fractional order derivatives.A straightforward method was used to determine the electromagnetic energy density distribution in a dispersive and dissipative met material made up of wire arrays and split-ring resonators [8].The paper demonstrated that the system's energy may be thought of as being made up of the energy densities of the electric and magnetic fields as well as energy densities associated with the medium's reaction.Therefore, the system's equations of motion for polarization have been compared with the corresponding differential equations of suitable RLC circuits to develop formulas for the energy density of the medium.The electrical RLC circuit with a fractional DE is investigated in [9], where a new auxiliary parameter was added to maintain the three physical measures C, L, and R's three dimensions.Through the circuit's physical properties of RLC, the analytical solution was specified in terms of Mittag-Leffler's formula.An analytical approach to examine the impacts of contemporary fractional differentiation on the RLC electrical circuit was discussed in [10].The leading RLC electrical circuit DE has been fractionalized with three kinds of partial derivatives, where the RLC electrical circuit was seen for unit step sources, periodic, and exponential.
Melnikov's theory for implied ordinary DEs of little amplitude perturbations was presented in [11].If certain Melnikov-like criteria are met, it is specifically explored how long orbits linking singularities can endure in finite time for nonlinear RLC circuit systems.The two delays in the Van der Pol delay model and delay differential equations were produced from ordinary differential equations using the Taylor series to define the model for treating Parkinson's disease [1].Floquet's theory to examine the linear dynamic analysis of RLC circuits representing AC generators with periodically time-varying inductances was presented in [12].The dynamic stability's prerequisites are derived.The stability domains and transition curves were predicted using the harmonic balancing approach.A fractional order differential equation can be used to describe the dynamics of noninteger order RLC electrical circuits by adding an auxiliary parameter [13], where an analytical determination of the filter parameters and numerical validation of the outcomes were used.Despite the fact that differential equations are marketed as applied mathematics, the course rarely includes any practical applications [14].Deep learning convolutional neural networks (DLCNN) [15]- [18] and reinforcement learning [19], [20] that were undergone fast growth in modern decades, were employed to solve the nonlinear DEs in general.Deep learning-based data assimilation algorithms [21], [22] were lately presented to train Navier-Stokes network formulas to estimate different quantities of interesting.DLCNN-based methods can be classified into three kinds: i) DLCNN that maps indirectly to the parameters or inner results of an algebraic explanation to use them for deriving numerical solutions [23], [24]; ii) DLCNN that maps directly toward the solutions with a discrete method, which is same as numerical solutions [25], [26]; and iii) DLCNN that maps straight to the solution characterized by a DLCNN with a continuous way and it is similar to that in analytical solutions [27]- [29].In this type, the data applied for training the network are arbitrarily modeled inside the whole solution range in every training batch, including boundary and initial circumstances.The necessary characteristic of all of these techniques is the adopting the benefit of the nonlinearity depiction capability of DLCNNs.New development [30] in mechanics with this capability has been stated such as, Li et al. [31] presented a generative adversarial model network for mapping hidden

METHOD 2.1. Object and research hypothesis
This work solves the RLC circuit differential equations by using Laplace transform.It is possible to define the Laplace transform by a function (), which is given by: ()   Calculations are kept in their native symbolic form rather than in numerical form using symbolic workflows.This method enables using precise symbolic values and comprehends the properties of the obtained answer.When a quantitative result is required or symbolically continuing is impossible, numbers are substituted for symbolic variables.In general, the symbolic workflows of solving equations include; equations definition, equations solving, values substitution, plotting results, and analyzing results.

Equations definition
To solve differential equations using beginning conditions, utilize the Laplace transform.We will consider solving the RLC circuit shown in Figure 1.Capacitor charge is referred to as () in coulomb, AC voltage source in volts is referred to as (), capacitance in farad is referred to as , inductance in henry is referred to as , currents in ampere are referred as I1, I2, I3, resistances in ohm are referred as R1, R2, R3.Apply Kirchhoff's current and voltage laws, following equations, are obtained:

Deep learning convolutional neural network framework
From a physical perspective, the initial and boundary circumstances are continued with time, and as a result, the parameters of the approach circuit are also continuous in time: the parameters at the current step are used as an ideal initialization for the next step, allowing the parameters of the DLCNN to transmit with the time phase in the solution procedure, which makes the process rapid for the remaining time stages.The derivatives of the current and capacitor charge equations are computed according to the gradient of the network output according to the input.The hidden layers activation functions are selected as rectified linear unit (ReLU) functions to conserve the stability of those derivatives.The network architecture is trained according to the computed loss based on the gradient descent process.A common nonlinear differential equation has the following format: The term (, ) represents the underlying equation solution, u represents the time derivative, while (, ) denotes the nonlinear function with parameter .Specifying the required condition, represented by the initial and boundary parameters, the equation solution is a nonlinear map within the solution range (, ).DLCNNs have revealed noteworthy achievements in the learning of nonlinear high-dimensional functions [32].The proposed DLCNN technique approximates the nonlinearity mapping that has a probabilistic network  (|) employed through a DLCNN with the parameter .The contender solution is sampled from the network probability, which means ̂∼  (|).The network's loss function for continuous solution action domain.The symbols employed here are recognizable from traditional DLCNN learning for simplicity, and Figure 2 shows how they apply to a particular solution problem.This research uses a DLCNN with six convolution layers and one fully linked layer.A max pooling layer, activation layer (linear rectified unit, or ReLU), and normalization batch layer are present in every convolution layer, with the exception of the final convolution level, when an average pooling level is utilized in place of the maximum layer.In the output layer, softmax activation is present.Figure 3 displays the screenshot for the specifics of the created network layers.The training configuration are: i) every nine epochs, the learning rate is decreased by a factor of 10; ii) setting the initial learning rate to 2e-2; iii) setting the maximum number of epochs to 12; iv) using a stochastic gradient descent with momentum (SGDM) solver with a mini-batch size of 256; and v) plotting the training progress.1.The charge and current initially decline exponentially.On the other hand, they oscillate over a long period of time.The terms "steady state" and "transient" are used to obtain these characteristics.The qualities of the result can be examined with a symbolic result, but this is not possible with a numeric result.Figure 4 demonstrated that examining  and 1 visually shows that these equations consist of many terms.The plotting of the behaviors over [0 15] is performed to determine their contributions.The plots in Figure 5 show that  has two steady-state terms and a transient, while I1sol has a steady-state term and transient.From visual inspection, it is noted that  and 1 are with an exp function term, which is assumed to cause the decay in transient exponentially.The Laplace transform and deep learning CNN solutions show excellent consistency for the steady solutions and there is a good agreement between the obtained transients.Future work can be recommended when comparing the solving of such symbolic differential equations of the 2-loop RLC AC circuit with deep learning convolutional neural network [33], [34].4. CONCLUSION This work solves the DEs of a two loops RLC circuit of an alternating voltage source by using Laplace transform.The capacitor charge and current initially decline exponentially and they oscillate over a long period.The qualities of the transient and steady-state results can be examined with a symbolic result, which is not possible with a numeric result.Three major points can be concluded: i) the approach presented has successfully modeled and solved the symbolic differential equations of the currents and the capacitor charging of the 2-loop RLC AC circuit; ii) the visualization of the obtained results of the capacitor charging and inductor current shows that these equations consist of many terms that are assumed to cause the decay in transient exponentially; and iii) this work presented the DLCNN-based solution and compared the results with its corresponding Laplace transform solution.

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ISSN:2302-9285 Bulletin of Electr Eng & Inf, Vol. 13, No. 1, February 2024: 490-498 492 variables for a microstructure to use in fabrication.The major contribution of this work is to model the DEs of a two loops RLC circuit of an alternating voltage AC-source by using two alternative approaches; Laplace transform and DLCNN in MATLAB environment.To obtain this aim, several objectives are accomplished such as: i) to model and solve the DEs of a 2-loop RLC AC circuit; ii) to plot and analyze the obtained results of the capacitor charging and inductor current; and iii) to solve and compare the results with the DLCNN-based solution.

Figure 2 .
Figure 2. DLCNN framework for the DEs of the adopted two-loop RLC circuit

Figure 5 .
Figure 5.The transient and steady state terms of; (a) 1 and (b)

Figure 6 .
Figure 6.The comparison of transient and steady-state behavior for (a) the charge  and (b) the current 1 over two unlike time periods given by; (c) 2 ≤  ≤ 25 and (d) 0 ≤  ≤ 15 Bulletin of Electr Eng & Inf ISSN:2302-9285  Modeling two loops RLC circuit AC power source using symbolic arithmetic …(Inaam Rikan Hassan) 497