Applications of Linear Algebra to Electrical Engineering - Research Paper Example

Applications of Linear Algebra to Electrical Engineering Engineering is heavily influenced by the mathematics. Engineering problems can be solved by utilizing the principles of mathematics. Calculus, algebra, etc are the branches, which are utilized in the Engineering problems. We can say that the practical application of the mathematics can be truly seen in the engineering problems. The examples can be seen when derivation and integration are utilized to resolve the electronic and electrical circuits. Linear Algebra is also utilized to solve the problems. Matrices are used to find the simplest method of solving complex electrical circuits. A mesh is loop that has a single current flow path. In the circuits where there are different current paths and different voltages, the simplest method is to utilize the matrices to solve for the currents and voltages. Every electrical network must have three basic quantities current, voltage or potential difference and resistance. There could be impedance or reactance at the place of resistance but the two quantities voltages and current are the constant in every electrical network. Many scientists proposed theories shoe the relation between the voltage or potential difference, current and resistance but ohms law presented the best relation in between the three quantities. The writers ‘Shamieh Cathleen and Gordon McComb’ in the book ‘Electronic for Dummies’ writes “Ohm’s law is like a master key, unlocking the secrets to the electronic circuits” (Shamieh & McComb 67-98). The problem then arose about solving the complex electrical networks, which have more than one resistance, current, and voltages. At every loop, the current is different and at every node, the difference in voltage and current could be seen. To solve the matter the engineers and scientists relied upon utilizing the different mathematical methods and formulas but linear algebra has the right method to solve the complex electrical circuits and electrical networks. Kirchhoff’s laws presented by Gustav Kirchhoff in 1845 relate with the power conservation in the electrical circuits and have vast applications in electrical engineering. As we know the Ohm’s law states that V = IR (Shamieh & McComb 67-98), it is the simplest estimation to calculate the voltage or current into the circuit if the third variable is known. Kirchhoff’s voltage law states, “The algebraic sum of the voltages around any closed path in a circuit is identically zero for all time” (Drof & Svoboda 69). In this way, if we consider a mesh in which, there only a single loop, the method that represent the mesh current is used to the determine the current in each of the mesh. If we have a single loop consider the situation, it will be easier to evaluate the unknown quantity but it the circuit consists of several mesh circuits, the will be complex to evaluate an unknown currents in different mesh circuits. For this reason, Kirchhoff’s voltage law provided an easier method to apply it on every mesh of the circuit. It should be kept in mind the circuit contains resistances, and if a mesh has a resistor that the other mesh also has, the voltage must be the product of the resistance and the subtraction of analyzed current and other currents. If we have two or more than two mesh circuits, the matrix method is an easier method to solve for the different currents. Example Consider a circuit in which, the source voltage is Vs = 120V, having Six resistances. It can be seen that the circuit has three loops and three different current paths. Each mesh current has an circular path and resistors R2 is common between the mesh 1 and mesh 2. Similarly, resistor R4 is common between mesh 2 and mesh 3. As according to the law, the sum of voltage is equal to zero, then voltage across mesh 1 is zero, sum of voltages across mesh 2 is zero and sum of voltages across mesh three is zero. Voltages across mesh 1 is Vs, I1R1, (I1-I2) R2, I1R3, so according to the law; Mesh 1; Vs - I1R1- (I1- I2) R2 - I1R3 = 0 Mesh 2; - (I2-I3) R4 - (I2-I1) R2 = 0 Mesh 3; I3R5 – I3RL – (I3 – I2) R4 = 0 As the values of resistances are assumed to have 2Ω, 5Ω, 2Ω, 4Ω, 15Ω and 5Ω for the resistances R1, R2, R3, R4, R5 and RL respectively. These are the assumed values and theses can vary. Putting the values in the above equation, we get. Mesh 1; 9I1 – 5I2 = 0 Mesh 2; 5I1 + 9I2 – 4I3 = 0 Mesh 3; - 4I2 + 24I3 = 0 Converting the above equations into the matrices form, we get | 9 -5 0 | | I1 | |120 | |-5 9 -4 | | I2 | = | 0 | | 0 -4 24 | | I3 | | 0 | The matrices from must respect the basic ohms law V = IR, then the different methods can be utilized to solve the matrix form for the unknown current in the loops. The most appropriate method is to submit the values in the augmented matrices and then evaluate the reduced Echelon matrix. In this way, the augmented matrix form becomes | 9 -5 0 120 | |-5 9 -4 0 | | 0 -4 24 0 | And the reduces Echelon matrix becomes | 1 0 0 20 | | 0 1 0 12 | | 0 0 1 2 | The 1 in first three columns represents I1, I2 and I3 while the forth column represents the values for the currents I1, I2, I3 which are 20A, 12A and 2A respectively. In this way, linear algebra helped to provide a simpler and easier method to solve the complex electrical circuit. Having the knowledge of the supply voltage and the resistances, it is an easier method to evaluate for the unknown currents. The currents give way to evaluate the voltages. As by the evaluation of the law states, “The voltage across independent current sources can be found only after the mesh current variables are solved.” (Kumar & Suresh 151). In the similar manner, the Kirchhoff’s current law states, “the algebraic sum of currents at a node is zero” (Kumar & Suresh 51). The law is simpler as compared to the Kirchhoff’s voltage law. Nodes in a circuit represent junctions or the joining points in the circuit. As the current can enter or leave the node, the current entering a node have a similar direction and the current leaving the node have the different direction. In this way, the leaving currents are subtracted from the currents entering the node to prove the Kirchhoff’s current law, which would be true meaning of the algebraic sum. The figure below represents a node. In the above figure, two currents I1 and I2 are entering the node while I3 and I4 are leaving the node so the Kirchhoff’s current law can be represented as; I1 + I2 – I3 – I4 = 0 Liner Algebra contributed massively to provide a solution for the electrical networks. Some of the other application, where the liner algebra contributed much is Linear Transformation, finding impedance in the electrical networks, determining admittance in the electrical networks, determining response in a circuit, etc. The utilization of the linear algebra in determining the unknown current in a circuit makes it easier for the person to evaluate the correct value. Other branches of mathematics largely influence the different fields of engineering. I have understood the importance of the linear algebra in the electrical engineering. The main reason of choosing the Kirchhoff’s law in my discussion is that the Kirchhoff’s law is the best suited example of the utilization of linear algebra and I had studied the relation between the Kirchhoff’s law and the linear algebra during my session. I came to understand that the linear algebra would enhance the understanding of the electrical networks and I could relate it to my studies. It enhanced my knowledge about the mesh current analysis. Works Cited Dorf, Richard and Svoboda, James A. Introduction to Electric Circuits. John Wiley & Sons, Inc, New York, 2001. Kumar, Suresh and Suresh, Kumar K. S. Electric Circuits & Networks. Pearson Education, India, 2008. Shamieh, Cathleen and McComb, Gordon. Electronics for Dummies. John Wiley & Sons, Inc, New York, 2001. Read More