Use of Maple to Analytically Solve the Equations of an Electrical Circuit Containing a Resistor, Diodes and Voltage Generator

Corresponding Author: M’hamed El Aydi Department of Physics, Faculty of Sciences, U.C.D, EL Jadida, Morocco Email: elaydi58@gmail.com Abstract: The role of technology and the use of software in the educational process are growing in recent times. The use of software is essential especially if the analytical method available is too complicated for the students. In this study, we used the Maple software to deal with two physics problems, in the first problem we consider an electrical circuit containing a resistor and two diodes powered by a sinusoidal voltage generator and in the second problem we consider an electrical circuit containing a resistor and a diode powered by a saw tooth voltage generator. For each problem we use Maple software to determine the exact analytical solutions for the current flowing in the different branches of the electronic circuit, we derive analytical expressions for the terminal voltages of all the elements of the circuit, we calculate the dynamic resistances diodes of the circuit and we animate graphic representations to study the influence of certain parameters on the current and the voltages at the terminals of all the elements of the circuit. The analytical solutions proposed are all expressed as functions of the Lambert W function.


Introduction
Maple is a proprietary computer algebra software allowing to manipulate mathematical expressions symbolically and thus to make exact calculations. Maple is a computer environment for advanced mathematics including tools for algebra, referential equations, mathematical analysis, discrete mathematics, graphical and numerical calculation, etc.
The transcendent equation of current intensity through a diode driven by a voltage source through a serial resistor is usually solved by accepting approximations. Fjeldly et al. (1991) exploited an approximate analytical resolution technique combined a test function with a series of expansion. This method leads to a precise solution without requiring a lot of computing time.
The authors (Pimbley et al., 1992) used Newton's method provides an accurate solution for negative values of normalized tension, but the precision of the solution is less acceptable for very large values of the normalized tension. Moreover, this method induced a lot of computing time.
In the work published by (Banwell and Jayakumar, 2000), the authors used the LambertW function to express the exact analytical solution for the normalized form of the generalized diode equation. The researcher in (Vargas-Drechsler, 2005) derived the same exact analytical solution using the computer algebra software Maple (Eberhart, 2009).
In this application worksheet, we consider two problems:  In the first problem, we consider an electrical circuit containing a resistor and two diodes powered by a sinusoidal voltage generator " Fig. 1", we use Maple software to determine exact 1670 analytical solutions for the current flows through the different branches of the electronic circuit. Then we derive analytical expressions for the voltages at the terminals of all the elements of the circuit and we represent them graphically using the Maple software. Finally, we calculate the dynamical resistances the diodes in the circuit. The proposed analytical solutions are all expressed as functions of the Lambert W function  In the second problem: Maple is used to determine exact analytical solution for the current flows through the Electrical circuit containing a resistor and diode powered by a saw tooth voltage generator presented in Fig. 2 and to study the influence of four parameters involved

First Problem
Exact analytical solution in electronic circuit containing a resistor and two diodes powered by a sinusoidal voltage generator (Fig. 1).

Materials and Methods
In this section, we introduce the used materials an methods: LambertW function (Dence, 2013), Kirchhoff's current law, Kirchhoff's voltage law, saturation current, dynamical resistance, Maple software.

LambertW Function
The function Lambert W is the reciprocal of the function f(x) = xe x . Since the equation x.exp(x) = y has an infinite number of solutions for a non-zero complex value. The function Lambert W admits an infinite number of branches in the complex plane and only two of these branches are real: i) The first branch, also called the main branch, vanishes at 0, it is noted LambertW0 and is defined

Kirchhoff's Current Law
Kirchhoff's Current Law (KCL) states that the algebraic sum of currents entering a node (or a closed boundary) is zero.
Kirchhoff's Voltage Law (KVL) states that the algebraic sum of all voltages around a closed path (or loop is zero.).
The current through the diode is: (Eberhart, 2009) Modeling the Problem using Maple Software Diode, Is1 its saturation current D2: Diode, Is2 its saturation current I = IR: Current through the resistance I1: Current through the diode D1 I2: Current through the diode D2 q: Electron charge eta: Ideality factor of the diodes kB: Boltzmann constant T: The absolute temperature V: Sinusoidal voltage source VAB: Voltage across the resistance VBC: Voltage across the diodes Rd1: Dynamical resistance of the diode D1 Rd2: Dynamical resistance of the diode D2 P: Period of V ..  ..
The numerical values used are such that: Graphical Representation of the Voltage Across the Resistance and Voltage Across the Diodes Graphical Representation of the Current Flowing through the Resistor, the Diode1, the Diode2 and

Second Problem
We considered the electrical circuit containing a resistor and a diode powered by a saw tooth voltage generator Fig. 7, we want to determine the analytical expressions for the voltages at the terminals of all elements in the circuit and we are interested in studying the influence of the saturation current, the temperature and the ideality factor.

Influence of the Loading Resistance
Here, we present the influence of the loading resistance on V(t); VAB(t) and VBC(t).

Influence of Ideality Factor on V(t); VAB(t) and VBC(t)
> restart: > with(plots): .  In  the intensities of the currents IR, I1 and I2 are positive on [0, P/2] and they reach their maximums at the same time t = P/4  the intensities of the IR currents, I1 and I2 are zero on [P/2, P], this is the role played by the diodes Figure 5 shows that:  The intensities of the current I1 and I2 are negative and almost null [P/2, P], because the diodes are not ideal Figure 6 shows that: The graphical representations allowed us to manipulate commands of the Maple software and to deduce the following results: Figure 8 shows that:  The dynamic resistance of the diode decreases very quickly in the time interval] 0,0.001] then it remains zero Figure 9 shows:  The voltage across the generator is a straight segment  For 0 < t <t0, (0.006 < t0 <0.007), the voltage VBC coincides with V and that VAB is zero Figure 10 and 11 shows:  For 0 <t <P/2, the IR current in the resistor is zero. the current IR increases and i is positive for P/2 <t <P The influence of load resistance, saturation current, temperature and ideality factor:  The results of Fig. 12 to 23 are summarized in the following Table 1 General Discussion The use of the Maple software allowed us to solve transcendent equations, to analytically express the solutions according to the LAMBERT W functions, to represent the solutions graphically and to make animations. Important question: Why don't we teach the LAMBERT W function and the Maple software at the secondary school?

Conclusion
We used the Maple software in several directions:  Modeling and problem solving, determination of exact analytical solutions of the expressions of current and voltage in the electronic circuit, Visualization of their evolutions as a function of time and illustration of the influence of certain parameters on the current and the voltages  This work is of interest to researchers in mathematics, physicists and teachers of mathematics or physics  The user of this article can make the following changes: i) Change the amplitude value Vm sinusoidal signal and watch the changes in graphs and animations ii) Change the physical parameters for example the temperature and look for the effect produced on the graphs and the animations iii) Substitute one signal for another and look at the influence on graphs and animations