APPLICATIONS OF LINEAR ALGEBRA IN ELECTRICAL ENGINERING Presented by: Shivansh Mehta Harshil Menpara Mojil Shah Mudra Makwana Yash Nakrani Sandip Nandaniya Navin Khatri Netra Ray Divyansh Lokwani

WHAT IS LINEAR ALGEBRA? Linear lagebra is the branch of mathematics concerning linear equations, linear functions such as and their representations through matrices and vector spaces. It is central to almost all areas of mathematics and is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis may be basically viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and engineering areas, because it allows modeling many natural phenomena, and efficiently computing with such models.

Loop current analysis of electric circuits In this method, we set up and solve a system of equations in which the unknowns are loop currents. The currents in the various branches of the circuit are then easily determined from the loop currents. The steps in the loop current method are: Count the number of required loop currents required. Call this number m. Choose m independent loop currents, call them I1, I2, . . . , Im and draw them on the circuit diagram.

Write down Kirchoff’s Voltage Law for each loop Write down Kirchoff’s Voltage Law for each loop. The result, after simplification, is a system of n linear equations in the n unknown loop currents in this form: where R11, R12, . . . , Rmm and V1, V2, . . . , Vm are constants.

Solve the system of equations for the m loop currents I1, I2, Solve the system of equations for the m loop currents I1, I2, . . . , Im using Gaussian elimination. Reconstruct the branch currents from the loop currents.

EXAMPLE: Find the current flowing in each branch of this circuit.

The number of loop currents required is 3. Solution: The number of loop currents required is 3. Therefore, we choose the loop currents shown in the diagram.

Write down Kirchoff’s Voltage Law for each loop Write down Kirchoff’s Voltage Law for each loop. The result is the following system of equations:

Collecting terms this becomes: Solving the system of equations using Gaussian elimination gives following currents, all measured in amperes: I1 = - 4.57, I2 = 13.7 and I3 = - 1.05

Reconstructing the branch currents from the loop currents gives the results as shown.

Wheatstone Bridge To illustrate the problem, we consider a generalised bridge circuit as shown below.

Pick closed loops called mesh currents. Apply Kirchhoff Voltage Law to each loop, being very careful with the sign convention to give a set of simultaneous equations.

Solve the simultaneous equations to find the loop currents and then the current through each component and the voltage across each component. Z I = V I = Z-1 V Therefore to solve the simultaneous equation we simply have to multiply the vector V by the inverse of the matrix Z.

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