Assignment on Linear Algebra

Question: Discuss about the Assignment on Linear Algebra. Answer: The problem: One of the most important application of Linear Algebra is calculating the voltage and currents in closed loop circuits, and this importance rises when calculating largely closed loops with a large number of electronic components. In electrical circuit analysis, a set of linear equations are used to analyze the direct flow of current (DC), and the coefficients of these equations are real because its direct current circuits. Also, the linear equations can be used to calculate the current and voltages in alternative closed loop electronic circuits but with real and imaginary coefficients for these linear equations [1]. This would save huge effort and a great amount of time for electronic circuit designers, especially when dealing with too complicated closed-loop circuits. Moreover, electronic circuit builders use various values of resistors in the circuit to get the ideal current flow for each loop by using electronic circuit simulation software. Alternatively, they also can use system linear equations to specify the current for each loop in the circuit by changing the coefficient values of the variables in the system linear equations. In this application, we will try to calculate the three currents running in the three loops which are very simple electronic circuit but could give a general idea in how to tackle large-scale circuits by following the same solution procedure. In the figure below we have three circuit loops and each loop will present a linear equation assuming that the three currents are running in each loop in clockwise direction. Solution In order to solve the problem of these electronic circuits, first I have to make linear equation from the given data. As per Kirchhoffs Voltage law. The sum of all voltage meeting in loop is equal to zero according this we will go loop by loop to frame it is linear equation. For Loop 1 5I1+10(I1-I2) +10+10(I1-I3)-10=0 For Loop 2 10I2+5+10(I2-I3)-10+10(I2-I1) =0 For Loop 3 10I2+5+10(I2-I3)-10+10(I2-I1) =0 If we simply the equation, we can see that it is just like linear equation in form f( x+y+z) = f(x) + f(y) + f(z) = c.f(x) where x, y, z are the variable and C =constant, in this condition we have to find x which is in the form current, Suppose it is I1, I2, I3. We have to frame the given coefficients in the form of linear algebra. For loop 1 25I1-10I2-10I3=0 For Loop 2 -10I1+30I2-10I3=5 For loop 3 -10I1-10I2+33I3=0 This is in the form of equation Ax=b Where, A = coefficient of the three currents for the three loops = b= the constant number after equality sign in our equations= x= the unknown three currents running the three loops= All the coefficients is framed in to solve this equation as a linear algebra, Putting all the values in mat lab, we can calculate the value of x i.e. all the current I1, I2, I3 by framing this into inverse of linear, matrix x =A-1*b Finally we get the result as given below x= Where I1 = 0.1541 Amp, I2 = 0.2599 Amp, and I3 = 0.1254 Amp Here we can see that what a linear equation can do in electronic circuits. We have taken the help of one computational advance techniques in engineering and that is MATLAB, The code of the MATLAB is given below clc; % constant number after the equality sign of the three simplified equation. b= [0;5;0]; % the coefficients of the currents for the three equations. A= [25 -10 -10; -10 30 -10; -10 -10 33]; % calculating the three currents. I=inv(A)*b; disp(I) References S. Dianat and E. Saber,Advanced linear algebra for engineers with MATLAB. Boca Raton, FL: CRC Press, 2009. J. Manassah,Elementary mathematical and computational tools for electrical and computer engineers using MATLAB. Boca Raton, FL: CRC/Taylor Francis, 2007. H. Anton,Elementary linear algebra. New York: John Wiley, 1994. D. Lay,Linear algebra and its applications. Boston: Pearson/Addison-Wesley, 2006.