Modeling Electric Current in a Two-Loop Circuit

Study Case Modeling Electric Current in a Two-Loop Circuit Using Differential Equations Michael Kurniawan (2018123456) University of Cincinnati, 2600 Clifton Ave, Cincinnati, OH 45221, United States kurniaml@mail.uc.edu Abstract Differential equations are used to model various natural phenomena involving changes in a quantity over time, one of which is the dynamics of electric current. The amount of electric charge flowing at a given time can be determined by solving differential equations. In a closed circuit (loop), electric current analysis follows Kirchhoff's Law. To determine the amount of charge flowing at time t, the solution of an nth-order differential equation is used. If the equation obtained is in the form of a nonhomogeneous differential equation with constant coefficients, it can be solved using the method of undetermined coefficients. To visualize changes in current strength over time, simulations can be performed using software such as Maple or MATLAB. Keywords: Differential Equations, Electric Current, Kirchhoff's Law. INTRODUCTION In electronics, the term electric current refers to the flow of charge in a closed circuit. A closed circuit consists of a voltage source and a load connected by a conductor, allowing electric current to flow [1]. Understanding the theory of electrical circuits is very important because this concept is widely applied in everyday life [2]. The theory of electrical circuits is rooted in the basic laws of physics developed by Coulomb (1785), Ohm (1827), Faraday (1831), and Kirchhoff (1845) [3]. In conductors, there are free charges in the form of positive and negative ions that move easily. When a conductor is subjected to an electric field, these charges move to form an electric current. An electrical circuit usually consists of several interconnected elements, with at MICHAEL KURNIAWAN 1 Study Case least one voltage or current source [4]. In a simple direct current (DC) system, Kirchhoff's Law, introduced by German physicist Gustav Robert Kirchhoff, is well known. He formulated two basic laws: Kirchhoff's Current Law (KCL) and Kirchhoff's Voltage Law (KVL) [5]. Previous studies have examined electrical circuit models using a mathematical approach. For example, Yuniarsi (2011) applied a numerical method to determine the solution of an electric current model in a closed circuit [6]. Al Firman (2015) used graph theory in the form of a spanning tree to determine the total resistance of a circuit [7]. Silmi and Rina (2017) applied the Gauss Elimination method using MATLAB to solve a mathematical model of electric current [8]. In addition to numerical approaches and graph theory, mathematical models can also be formulated in the form of differential equations [9]. Differential equations involve independent variables and their derivatives with respect to independent variables, and solutions are obtained through specific methods according to the general form of the equation. One popular method is the undetermined coefficient method [10]. With this method, the electric current flowing at time t can be calculated by solving differential equations with certain initial conditions. Therefore, this study discusses the mathematical model of electric current in a closed circuit using the method of undetermined coefficients differential equation. RESEARCH METHOD This study uses a qualitative descriptive approach, focusing on physical phenomena that are often encountered in everyday life, namely the flow of electric current in a closed circuit. The author formulates this phenomenon into a mathematical model based on differential equations, which are then solved using the undetermined coefficient method. The mathematical solutions obtained are then interpreted to describe the actual electrical current phenomenon. The overall research process flow can be seen in Figure 1. MICHAEL KURNIAWAN 2 Study Case Figure 1. Simple circuit diagram based on Kirchhoff's First Law. This research also requires a theoretical basis covering: • Electric current • Kirchhoff's Law • Differential equations • The method of undetermined coefficients 2.1. Electric Current Electric current is the movement of electric charge in a conductor. Current is measured in Coulombs per second (C/s) or Amperes (A), while electric charge is measured in Coulombs (C). If the current is constant, the current I can be expressed as: 𝐼= 𝑄 𝑡 (1) with: I: electric current (A) Q: electric charge (C) t: time (s) The integral form is: 𝑄 = ∫ 𝐼 𝑑𝑡 (2) In general, the current flowing at time t can be determined using the derivative with respect to time: 𝐼= MICHAEL KURNIAWAN 𝑑𝑄 𝑑𝑡 (3) 3 Study Case 2.2. Kirchhoff's Laws Kirchhoff's Laws consist of two parts: Kirchhoff's Current Law (KCL) and Kirchhoff's Voltage Law (KVL). Law I (KCL) states that the sum of the currents entering a junction point is equal to the sum of the currents leaving that point. For example, for a junction point, the following applies: 𝑖1 + 𝑖2 + 𝑖3 = 𝑖4 + 𝑖5 Figure 2. A simple circuit illustrating Kirchhoff's First Law. In general, it can be written as: (4) ∑𝐼 = 0 Meanwhile, Kirchhoff's Second Law (KVL) states that the algebraic sum of the voltages along a closed path in a circuit is zero. In other words: ∑𝑉 = ∑𝐼 ⋅ 𝑅 = 0 (5) In its application, if the direction of the current is the same as the direction of the loop, then the value is considered positive; conversely, if the directions are opposite, the value is negative. Similarly, if the positive pole of the voltage source is encountered first, then the electromotive force (EMF) is positive, and conversely, it is negative [11]. Table 1. Voltage Drop in Various Electrical Circuit Elements 2.3. Differential Equations The general form of an nth-order linear differential equation is: 𝑎𝑛 𝑦 (𝑛) + 𝑎𝑛−1 𝑦 (𝑛−1) + 𝑎𝑛−2 𝑦 (𝑛−2) + ⋯ + 𝑎1 𝑦 ′ + 𝑎0 𝑦 = 𝑔(𝑥) The general solution to this equation is expressed as: 𝑦 = 𝑦𝑐 + 𝑦𝑝 MICHAEL KURNIAWAN 4 (7) (6) Study Case where: y_c: complementary solution (for g(x)=0) y_p: particular solution (for specific inhomogeneous conditions) 2.4. Method of Undetermined Coefficients This method is used to solve linear non-homogeneous differential equations with constant coefficients. The steps are as follows [10]: 1. Determine the complementary solution y_c. 2. Use the elimination operator on the function g(x) to obtain the general form of the solution. 3. Determine the particular solution y_p based on the form of g(x). 4. Combine the two to obtain the total solution y=y_c+y_p. RESULTS AND DISCUSSION Consider a closed circuit with two loops as shown in Figure 3, consisting of a resistor (R) in ohms, an inductor (L) in henries, a capacitor (C) in farads, and a voltage source (E) in volts. Figure Caption: • R: Resistance in ohms • C: Capacitor in farads • L: Inductor in henries • E0: Electromotive force (EMF) in volts • I: Electric current in amperes Figure 3. Electric current flow in a simple two-loop circuit. According to Kirchhoff's Second Law (Kirchhoff's Voltage Law), the algebraic sum of all voltages in a closed loop is zero. The voltage drop across the resistor, capacitor, MICHAEL KURNIAWAN 5 Study Case and inductor is VR=RI, V_C=Q/C, and VL=L dI/dt, respectively, where Q is the electric charge and E0 is the source voltage. Therefore, based on Kirchhoff's law for loop 1, we obtain: −𝐸0 + 𝑉𝐿 + 𝑉𝑅1 = 0 𝑑𝐼1 𝐸0 − 𝐿 − 𝑅1 (𝐼1 − 𝐼2 ) = 0 𝑑𝑡 𝑑𝐼1 𝐸0 𝑅1 𝑅1 = − 𝐼1 + 𝐼2 𝑑𝑡 𝐿 𝐿 𝐿 (8) Next, for loop 2, we have: 𝑉𝐶 − 𝑉𝑅1 + 𝑉𝑅2 = 0 𝑄 − 𝑅1 (𝐼1 − 𝐼2 ) + 𝑅2 𝐼2 = 0 𝐶 1 𝑑𝐼2 𝑑𝐼1 𝑑𝐼2 − 𝑅1 + (𝑅1 + 𝑅2 ) =0 𝐶 𝑑𝑡 𝑑𝑡 𝑑𝑡 (9) Substituting Equation (8) into Equation (9), we obtain: 𝑑𝐼2 𝑅1 𝐸0 𝑅12 𝑅1 𝑅2 1 = − 𝐼1 + 𝐼2 − 𝐼 𝑑𝑡 (𝑅1 + 𝑅2 )𝐿 (𝑅1 + 𝑅2 )𝐿 (𝑅1 + 𝑅2 )𝐿 𝐶(𝑅1 + 𝑅2 ) 2 (10) By differentiating the first equation with respect to time, we obtain: 𝑑𝐼1 = 50 − 25𝐼1 + 25𝐼2 𝑑𝑡 𝑑𝐼2 100 50 = − 𝐼1 + 15𝐼2 𝑑𝑡 3 3 (11) Then, substituting Equation (11) into Equation (12) yields: 𝑑2 𝐼1 𝑑𝐼1 125 250 + 10 + 𝐼1 = 2 𝑑𝑡 𝑑𝑡 3 3 (13) Equation (13) is a non-homogeneous differential equation with constant coefficients, which can be solved using the method of undetermined coefficients. The first step is to determine the characteristic equation: 𝜆2 + 10𝜆 + 125 =0 3 (14) The following eigenvalues are obtained: 𝜆1 = −5 + 5√6 5√6 𝑖, 𝜆2 = −5 − 𝑖 3 3 (15) Thus, the complementary solution is: 𝐼𝑐 (𝑡) = 𝑐1 𝑒 (−5+ 5√6 𝑖)𝑡 3 + 𝑐2 𝑒 (−5− The forcing function is: MICHAEL KURNIAWAN 6 5√6 𝑖)𝑡 3 (16) Study Case 𝑔(𝑡) = 250 3 (17) Therefore, the general solution can be written as: 5√6 5√6 𝐼1 (𝑡) = (𝑐1 + 𝑐2 )𝑒 −5𝑡 cos⁡( 𝑡) + (𝑐1 − 𝑐2 )𝑖𝑒 −5𝑡 sin⁡( 𝑡) + 2 3 3 (18) Substituting I1 (t) into system (11) yields: where: 5√6 5√6 𝐼2 (𝑡) = 𝑐3 𝑒 −5𝑡 cos⁡( 𝑡) + 𝑐4 𝑒 −5𝑡 sin⁡( 𝑡) 3 3 𝑐3 = (19) 4 15 (𝑐1 + 𝑐2 ) + (𝑐1 − 𝑐2 ) 5 √6𝑖 𝑐4 = 4 15 (𝑐1 − 𝑐2 ) − (𝑐1 + 𝑐2 ) 5𝑖 √6 (20) If the initial conditions I1 (0)=0 and I2 (0)=2 are used, then the system can be simulated using Maple or MATLAB software. The plot results are shown in Figure 4. Figure 4. Simulation of changes in electric current over time. Color Legend: • Red: Changes in I1 (t) over time • Blue: Change in I2 (t) over time From Figure 4, it can be seen that the current I1 at the initial time (t=0) is zero because there is no charge flowing, then it increases and finally stabilizes at a value of around 2 Amperes. Conversely, the current I2 initially has a value of 2 Amperes and gradually decreases to near zero. This phenomenon is consistent with Kirchhoff's Law, where the sum of the currents entering is equal to the sum of the currents leaving at a branch point. In other words, the MICHAEL KURNIAWAN 7 Study Case total electric charge flowing in a closed circuit is constant, even though the direction of the current may change over time. At some intervals, the direction of current I2 is opposite to the main loop, so its value becomes negative, as seen in the graph when t≈0.8 seconds. CONCLUSION The amount of electric current flowing in a circuit can be determined using differential equations. In a two-loop closed circuit (Figure 3), the differential system that describes the dynamics of the current is: 𝑑𝐼1 = 50 − 25𝐼1 + 25𝐼2 𝑑𝑡 𝑑𝐼2 100 50 = − 𝐼1 + 15𝐼2 𝑑𝑡 3 3 (21) The general solution is: 𝐼1 (𝑡) = (𝑐1 + 𝑐2 )𝑒 −5𝑡 cos⁡( 𝐼2 (𝑡) = 𝑐3 𝑒 −5𝑡 cos⁡( 5√6 5√6 𝑡) + (𝑐1 − 𝑐2 )𝑖𝑒 −5𝑡 sin⁡( 𝑡) + 2 3 3 5√6 5√6 𝑡) + 𝑐4 𝑒 −5𝑡 sin⁡( 𝑡) 3 3 (22) where: 4 15 4 15 𝑐3 = (𝑐1 + 𝑐2 ) + (𝑐1 − 𝑐2 ), 𝑐4 = (𝑐1 − 𝑐2 ) − (𝑐1 + 𝑐2 ) 5 5𝑖 √6𝑖 √6 (23) The simulation results show that the currents in loop 1 (I1) and loop 2 (I2) have an inverse relationship with time. When I1 increases, I2 decreases, and vice versa. Over time, both reach a steady state. This pattern illustrates the balance of energy and charge in accordance with the basic principles of Kirchhoff's Law. MICHAEL KURNIAWAN 8 Study Case REFERENCE MICHAEL KURNIAWAN 9