Analysis of Complex Electrical Circuits

Study Case Analysis of Complex Electrical Circuits Using Linear Equation Systems and LU Decomposition Methods Assisted by MATLAB Michael Kurniawan (2018123456) University of Cincinnati, 2600 Clifton Ave, Cincinnati, OH 45221, United States kurniaml@mail.uc.edu Abstract This study discusses the comparison between simulation results and physical experiments in three-loop mesh circuit analysis using the Proto Circuit Android App. Linear equation systems are an important topic in algebra with various applications in science and engineering. One such application is in electrical engineering, particularly in electrical circuit analysis. Linear equation systems in electrical circuits are used to determine the relationship between current and voltage in complex circuits. Through the application of Kirchhoff's law, a system of equations can be formed that represents each element in the circuit. The system of equations is then solved using the LU decomposition method, which is a technique that breaks a matrix into two triangular matrices: the lower triangular matrix (𝐿) and the upper triangular matrix (𝑈). In this study, the Doolittle method was used to perform matrix decomposition. The system of equations was solved using MATLAB R2019b software, which automates the calculation process, enabling faster and more accurate analysis of various circuit configurations. Based on the analysis results, the current values in each branch of the circuit were obtained, namely: 𝐼₁ = 11.4286 A, 𝐼₂ = 9.2857 A, and 𝐼₃ = 2.1429 A. These values represent the amount of current flowing through each component in the circuit. Keywords: Linear Equation System; Electrical Circuit; Kirchhoff's Law; LU Decomposition; MATLAB. MICHAEL KURNIAWAN 1 Study Case INTRODUCTION A linear equation system is a set of equations, each of which is linear with respect to its variables. The main objective of solving this system is to determine the values of the variables that satisfy all the given equations simultaneously. There are various methods for solving linear equation systems, including substitution, Gauss–Jordan elimination, and LU decomposition. Linear equation system models are commonly found in various disciplines, both in theoretical contexts and in solving real-world problems [1–6]. In electrical engineering, the concept of linear equation systems is applied to analyze electrical circuits. This analysis is based on Kirchhoff's Laws [7], which consist of the current law (KCL) and the voltage law (KVL). For circuits with a single loop, only one equation is formed. However, in circuits with multiple loops, a number of interrelated equations are formed—known as multivariable linear equation systems. One effective method for solving such systems is the LU decomposition method. This method has proven to be efficient in solving various numerical problems and has been widely used in engineering and computational science [8, 9]. For systems of equations with two or three variables, calculations can be done manually. However, for systems with a large number of variables (n > 3), manual calculations become inefficient and prone to errors. Therefore, this study uses MATLAB R2019b software to implement the LU decomposition method. The results show that the use of MATLAB can speed up the solution process and improve the accuracy of calculations in complex electrical circuit analysis. METHOD This study uses a quantitative approach with an experimental method, which is applied to a case study of electrical circuit analysis. Problems in the circuit are solved using the LU decomposition method. To convert electrical circuits into linear equation systems, Kirchhoff's Law is used as the basis for the formulation. Linear Equation System A linear system consisting of m equations with n unknown variables x_1,x_2,…,x_n can be written as follows [10, p. 272]: MICHAEL KURNIAWAN 2 Study Case 𝑎11 𝑥1 + 𝑎12 𝑥2 + ⋯ + 𝑎1𝑛 𝑥𝑛 𝑎21 𝑥1 + 𝑎22 𝑥2 + ⋯ + 𝑎2𝑛 𝑥𝑛 = 𝑏1 = 𝑏2 ⋮ = 𝑏𝑚 𝑎𝑚1 𝑥1 + 𝑎𝑚2 𝑥2 + ⋯ + 𝑎𝑚𝑛 𝑥𝑛 The above system is called linear because each variable x_k has an exponent of one. The value a_ij is the coefficient of the system, while b_i is the constant on the right side. In matrix form, the system of equations can be written as: Ax=b where A=[a_jk] is an m×n coefficient matrix, x^T=[x_1,x_2,…,x_n], and b^T=[b_1,b_2,…,b_m]. LU Decomposition One efficient method for solving linear equation systems is the LU decomposition method. This method factors the matrix A into the product of two matrices, namely the lower triangular matrix (L) and the upper triangular matrix (U), so that: A=LU The general form of these matrices is: 𝑎11 𝑎21 [ 𝑎31 ⋮ 𝑎𝑛1 𝑎12 𝑎22 𝑎32 ⋮ 𝑎𝑛2 𝑎13 𝑎23 𝑎33 ⋮ 𝑎𝑛3 ⋯ ⋯ ⋯ ⋱ ⋯ 𝑎1𝑛 1 𝑎2𝑛 𝑚21 𝑎3𝑛 ] = [𝑚31 ⋮ ⋮ 𝑎𝑛𝑛 𝑚𝑛1 0 1 𝑚32 ⋮ 𝑚𝑛2 0 0 1 ⋮ 𝑚𝑛3 ⋯ ⋯ ⋯ ⋱ ⋯ 0 𝑢11 0 0 0][ 0 0 ⋮ 1 0 𝑢12 𝑢22 0 ⋮ 0 𝑢13 𝑢23 𝑢33 ⋮ 0 ⋯ ⋯ ⋯ ⋱ ⋯ 𝑢1𝑛 𝑢2𝑛 𝑢3𝑛 ] ⋮ 𝑢𝑛𝑛 According to Howard [11, p. 492], the steps for solving a system of linear equations using the LU decomposition method are as follows: 1. Convert the equation Ax=b to LUx=b. 2. Suppose Ux=y, then the system becomes Ly=b. 3. Solve the system Ly=b to obtain the value of y. 4. Substitute the result y into Ux=y to obtain the value of x. Factorize the matrix A into L and U using the Doolittle method, using the formula [Erwin, p. 854]: 𝑢1𝑘 = 𝑎1𝑘 , 𝑘 = 1, … , 𝑛 𝑎𝑗1 𝑚𝑗1 = , 𝑗 = 2, … , 𝑛 𝑢11 𝑗−1 𝑢𝑗𝑘 = 𝑎𝑗𝑘 − ∑ 𝑚𝑗𝑠 𝑢𝑠𝑘 , 𝑘 = 𝑗, … , 𝑛 𝑠=1 MICHAEL KURNIAWAN 3 Study Case 𝑘−1 𝑚𝑗𝑘 1 = (𝑎 − ∑ 𝑢𝑘𝑘 𝑗𝑘 𝑚𝑗𝑠 𝑢𝑠𝑘 ), 𝑗 = 𝑘 + 1, … , 𝑛 𝑠=1 Kirchhoff's Law Kirchhoff's Law is the basis for electrical circuit analysis, which explains the relationship between current and voltage in an electrical network [12, pp. 53–54]. 1. Kirchhoff's First Law (KCL – Kirchhoff's Current Law) The algebraic sum of currents (𝐼) entering a point is equal to the sum of currents leaving that point: ∑I_in =∑I_out 2. Kirchhoff's Second Law (KVL – Kirchhoff's Voltage Law) The algebraic sum of the product of current (𝐼) and resistance (𝑅) in all elements in a closed loop plus the sum of electromotive forces (𝜀) is equal to zero: ∑IR+∑ε=0 The steps for solving electrical circuit analysis using the LU decomposition method are as follows: 1. Convert the electrical circuit using Kirchhoff's law into a system of linear equations. 2. Solve the system of equations using the LU decomposition – Doolittle method. 3. Determine the current values flowing in each branch of the circuit. This study uses MATLAB R2019b as a computational tool. The pseudocode of the LU decomposition method in solving electrical circuit problems is shown in Table 1 below. Table 1. LU Decomposition Method in Solving Electrical Circuit Problems MICHAEL KURNIAWAN 4 Study Case RESULTS AND DISCUSSION The analysis was performed on the electrical circuit shown in Figure 1. Figure 1. Electrical Circuit 𝐼1 𝐼2 2Ω𝐼3 4Ω40𝑉8Ω20𝑉 Based on Table 1 and the application of Kirchhoff's Law, the following system of linear equations is obtained: 𝐼1 − 𝐼2 − 𝐼3 = 0 2𝐼1 + 8𝐼3 = 40 4𝐼2 − 8𝐼3 = 20 This system of equations was solved using the LU decomposition method with MATLAB R2019b, and the calculation results are shown in Figure 2 below. Figure 2. MATLAB Computation Results 𝐼1 = 11.4286 A, 𝐼2 = 9.2857 A, 𝐼3 = 2.1429 A These results show that the current flowing in each branch complies with Kirchhoff's law and Ohm's law. The solution using LU decomposition proved to be efficient because it avoided the iterative calculations that usually occur in the traditional Gauss elimination method. MICHAEL KURNIAWAN 5 Study Case CONCLUSION In this study, the LU decomposition method was successfully applied to solve a system of linear equations derived from electrical circuit analysis. The coefficient matrix was factorized into two matrices, L and U, and then solved using MATLAB R2019b software. The results obtained were the values of electric current flowing through each component in the circuit, with a good level of accuracy. This method can be applied to more complex electrical circuit analysis, such as dynamic circuits and electrical network modeling. As a suggestion, further research could use other numerical methods, such as Cholesky decomposition or the Gauss-Seidel iterative method, to compare the speed and accuracy of the calculation results. REFERENCE MICHAEL KURNIAWAN 6 Study Case MICHAEL KURNIAWAN 7