Application of Cramer's Method for Electrical Circuit

Study Case Application of Cramer's Method for Electrical Circuit Analysis with Scilab Program Support Michael Kurniawan (2018123456) University of Cincinnati, 2600 Clifton Ave, Cincinnati, OH 45221, United States kurniaml@mail.uc.edu Abstract Various problems in electrical circuit analysis are generally solved using mathematical models. These models are then formulated numerically in the form of a Linear Equation System (LES). These systems can be solved using several numerical methods, such as the Gauss Elimination Method, Cramer's Method, Matrix Inverse Method, or Gauss– Jordan Elimination Method. To solve m equations with n unknown variables in large and complex systems, manual calculations would take a long time and be inefficient. Therefore, software such as Scilab can be used to facilitate fast and accurate mathematical calculations. This paper focuses on solving Linear Equation Systems using the Cramer Method implemented in Scilab. Keywords: Cramer Method, Electrical Circuit Analysis, Scilab. INTRODUCTION In various engineering and science disciplines, many problems can be modeled mathematically, one of which is in electrical circuit analysis. This type of analysis is usually formulated in the form of a mathematical equation system, which can consist of more than two equations and is often quite complex. Numerical methods combined with the help of computers enable the system to be solved quickly and accurately. As an illustration, a linear system of equations with m equations and n variables can be written as: 𝑎11 𝑥1 + 𝑎12 𝑥2 + ⋯ + 𝑎1𝑛 𝑥𝑛 𝑎21 𝑥1 + 𝑎22 𝑥2 + ⋯ + 𝑎2𝑛 𝑥𝑛 ⋮ 𝑎𝑚1 𝑥1 + 𝑎𝑚2 𝑥2 + ⋯ + 𝑎𝑚𝑛 𝑥𝑛 MICHAEL KURNIAWAN 1 = 𝑏1 = 𝑏2 = 𝑏𝑚 Study Case where a_ijand b_is are scalar numbers, where a_ijis called the coefficient, b_is called the constant, and x_1,x_2,…,x_nis the variables. This system can be represented in matrix multiplication form: [A][X]=[B] To solve this system, there are various numerical methods such as the Gauss Elimination Method, Cramer's Method, Matrix Inverse Method, and Gauss–Jordan Method. This paper focuses on the use of Cramer's Method in solving linear equation systems. For large and complex systems, the use of computers is very important. In this case, Scilab serves as an effective tool for solving SPLs using Cramer's method. THEORETICAL STUDY Basic Laws of Electricity 1. Ohm's Law When an electric current passes through a conductor, a potential difference arises between its two ends. Ohm's Law states that voltage (V) is directly proportional to the current (I) flowing through the conductor, with a constant factor of resistance (R). Mathematically: V=I×R 2. Kirchhoff's First Law (Kirchhoff's Current Law / KCL) The sum of the currents entering a branch point (node) is equal to the sum of the currents leaving that point. In other words, the algebraic sum of all currents at a node is zero. ∑𝐼 = 0 or ∑𝐼in = ∑𝐼out Example: 𝐼2 + 𝐼4 = 𝐼1 + 𝐼3 ⇒ 𝐼2 + 𝐼4 − 𝐼1 − 𝐼3 = 0 MICHAEL KURNIAWAN 2 Study Case 3. Kirchhoff's Second Law (Kirchhoff's Voltage Law / KVL) The sum of voltages in a closed loop is zero. This means that the algebraic sum of all voltages across the components forming the closed loop must be zero. 𝑉𝑎𝑏 + 𝑉𝑏𝑐 + 𝑉𝑐𝑑 + 𝑉𝑑𝑎 = 0 or −𝑉1 + 𝑉2 − 𝑉3 + 0 = 0 ⇒ 𝑉2 − 𝑉1 − 𝑉3 = 0 B. Cramer's Method A linear equation system consists of a number of equations with variables whose values are to be determined. There are two main approaches to solving them: • Direct methods, such as Gaussian elimination, Gauss–Jordan elimination, matrix inversion, and Cramer's method. • Indirect (iterative) methods, such as Jacobi Iteration and Gauss–Seidel Iteration. This article discusses the solution of linear systems using Cramer's Method, which is a method that uses matrix determinants to find the value of each variable. If the linear system is expressed as: [A][X]=[B] where A is an n×n square matrix, and B is an n×1 column matrix, then the solution for each variable can be obtained using the formula: 𝑋𝑗 = det⁡(𝐴𝑗 ) for 𝑗 = 1,2, … , 𝑛 det⁡(𝐴) where A_j is the matrix obtained by replacing the j^(th) column of A_j with matrix B, and det (A) ≠ 0. Example: Suppose the following system of linear equations: 𝑎11 𝑖1 + 𝑎12 𝑖2 + 𝑎13 𝑖3 𝑎21 𝑖1 + 𝑎22 𝑖2 + 𝑎23 𝑖3 𝑎31 𝑖1 + 𝑎32 𝑖2 + 𝑎33 𝑖3 = 𝑏1 = 𝑏2 = 𝑏3 Its matrix form: 𝑎11 𝑎21 𝑎31 𝑎12 𝑎22 𝑎32 𝑎13 𝑖1 𝑏1 𝑎23 ][𝑖2 ] = [𝑏2 ] 𝑎33 𝑖3 𝑏3 Then: MICHAEL KURNIAWAN 3 Study Case 𝑖1 = det⁡(𝐴1 ) det⁡(𝐴2 ) det⁡(𝐴3 ) , 𝑖2 = , 𝑖3 = det⁡(𝐴) det⁡(𝐴) det⁡(𝐴) C. Scilab Scilab is open source and free numerical computation and two- and threedimensional data visualization software for both personal and professional use. Scilab provides a high-level numerically oriented programming language and supports direct execution without compilation. Scilab can be run on various operating systems such as Windows, macOS, and Linux. DISCUSSION In solving numerical problems using computer programs, there are six main steps that must be taken, namely: 1. Modeling All parameters in the problem are converted into a mathematical model in the form of a system of equations. 2. Numerical Formulation After the mathematical model is constructed, the next step is to formulate it numerically and determine the numerical method to be used to obtain the solution. 3. Algorithm Development From the selected numerical method, an algorithm is created that represents the steps of the solution. 4. Programming The algorithm is then translated into a computer program. Typically, this stage begins with the creation of a flowchart before being implemented in code. 5. Operational The program is run using test data to ensure that the algorithm works correctly before being used on actual data. 6. Evaluation After the program is run with actual data, the results are interpreted. This stage includes analyzing the computational results and comparing them with basic MICHAEL KURNIAWAN 4 Study Case theory and empirical results to assess the accuracy of the numerical solution. Algorithm for Analyzing Electrical Circuits Using Cramer's Method in Scilab The steps of the algorithm for solving a system of linear equations using Cramer's method in the Scilab program are as follows: 1. Enter matrices A and B. 2. Check the size of both matrices — matrix A must be a square matrix, while matrix B must be a column vector with the same number of rows as matrix A. 3. Calculate the determinant value of matrix A. 4. Check whether the determinant of matrix A is not equal to zero. If det(A) = 0, then the system does not have a single solution (it may have many solutions or no solution at all). 5. Form matrices A₁, A₂, ..., Aₙ by replacing the jth column of A using matrix B, then calculate their determinants. 6. Determine the value of each variable by dividing the determinant of Aⱼ by det(A). Figure 1. Scilab Program Flowchart. MICHAEL KURNIAWAN 5 Study Case Case Study 1: Analysis of an Electrical Circuit with 3 Variables Figure 2. Three-Variable Electrical Circuit. Based on Kirchhoff's law, the following system of equations is obtained: 0𝑖1 + 8𝑖2 + 6𝑖3 0𝑖1 + 6𝑖2 + 9𝑖3 −𝑖1 + 𝑖2 + 𝑖3 = 12 =8 =0 This can be arranged in matrix form: [A][I]=[B] After the matrix is formed, the values of currents i_1, i_2, and i_3 are calculated using Cramer's rule: 𝑖𝑗 = det⁡(𝐴𝑗 ) det⁡(𝐴) Manual calculations yield: 𝑖1 𝑖2 𝑖3 −52 = 1.444 𝐴 −36 −60 = = 1.667 𝐴 −36 8 = = −0.222 𝐴 −36 = Using the Scilab program, the same results as the manual calculation are obtained: i₁ = 1.444 A, i₂ = 1.667 A, and i₃ = -0.222 A. Case Study 2: Analysis of an Electrical Circuit with 6 Variables Figure 3. Six-Variable Electrical Circuit. From the application of Kirchhoff's law, the following system of equations is obtained: MICHAEL KURNIAWAN 6 Study Case 76𝑖1 − 25𝑖2 − 50𝑖3 + 0𝑖4 + 0𝑖5 + 0𝑖6 −25𝑖1 + 56𝑖2 − 𝑖3 − 30𝑖4 + 0𝑖5 + 0𝑖6 −50𝑖1 − 𝑖2 + 106𝑖3 − 55𝑖4 + 160𝑖5 − 25𝑖6 0𝑖1 − 30𝑖2 − 55𝑖3 + 160𝑖4 − 25𝑖5 − 50𝑖6 0𝑖1 + 0𝑖2 + 0𝑖3 − 25𝑖4 + 56𝑖5 − 𝑖6 0𝑖1 + 0𝑖2 + 0𝑖3 − 50𝑖4 − 𝑖5 + 106𝑖6 = 10 =0 =0 =0 =0 =0 The system is expressed in matrix form: [A][I]=[B] Using Cramer's method, each current is calculated as: 𝑖𝑗 = det⁡(𝐴𝑗 ) det⁡(𝐴) The calculation results are obtained: 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 = 0.2706 𝐴 = 0.1748 𝐴 = 0.1239 𝐴 = 0.0966 𝐴 = 0.0439 𝐴 = 0.0460 𝐴 Calculations using Scilab show identical results to the manual method: i₁ = 0.2706 A; i₂ = 0.1748 A; i₃ = 0.1239 A; i₄ = 0.0966 A; i₅ = 0.0439 A; i₆ = 0.0460 A. Figure 4. Scilab Program for Solving a 6-Variable Circuit. MICHAEL KURNIAWAN 7 Study Case Figure 5. Display of Computation Results in Scilab. CONCLUSION Based on the discussion in the electrical circuit analysis case study, it can be concluded that Cramer's method is an effective approach for solving linear equation systems in electrical circuit analysis. The implementation of this method using Scilab provides accurate, fast, and efficient results, especially for complex and large-scale problems. Thus, Scilab has proven to be a very useful tool in the application of numerical methods in electrical circuit analysis. MICHAEL KURNIAWAN 8 Study Case REFERENCE MICHAEL KURNIAWAN 9