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# Application of linear algebra in ETE

Daffodil Internartional University

Daffodil Internartional University

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### Application of linear algebra in ETE

1. 1. WELCOME
2. 2. Presentation by: 1. Al-Amin Prince, ID: 141-19-1539 2. Nusrat Jahan ID: 141-19-1542 Department of ETE Daffodil International University. Guided By: Md. Mosfiqur Rahman Senior Lecturer, Department of General Educational Development Faculty of Science and Information Technology.
3. 3. Presentation on Application of Linear Algebra in ETE
4. 4. What is Linear Algebra? Linear Algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces. Hence, the above definition confirms that Linear Algebra is an integral part of mathematics.
5. 5. Applications of Linear Algebra in various fields Abstract Thinking Chemistry Coding Theory Cryptography Economics Elimination Theory Games Genetics Geometry Graph Theory Heat Distribution Image Compression Linear Programming Markov Chains Networking Sociology The Fibonacci Numbers Eigenfaces
6. 6. LINEAR ALGEBRA • Linear Algebra most apparently uses by electrical engineers. • When ever there is system of linear equation arises the concept of linear algebra. • Various electrical circuits solution like Kirchhoff's law , Ohm’s law are conceptually arise linear algebra. • To solve various linear equations we need to introduce the concept of linear algebra. • Using Gaussian Elimination not only computer engineers but most of daily computational work minimized . • Now we don’t have to use extremely large number of pages to calculate complex system of linear equations.
7. 7. GAUSSIAN ELIMINATION To fix all the assertion that we have performed earlier we use Gaussian elimination. In this method we need to keep all eqs. into matrix form, for e.g. Since the columns are of same variable it’s easy to do row operation to solve for the unknowns.
8. 8. This method is known as Gaussian Elimination. Now, for large circuits, this will still be a long process to row reduce to echelonform. With the help of a computer and the right software , the large circuits consisting of hundreds of thousands of components can be analyzed in a relatively short span of time. Today’s computers can perform billions of operations within a second, and with the developments in parallel processing, analyses of larger and larger electrical systems in a short time frame are very feasible
9. 9. THE WHEATSTONE BRIDGE The next application is a simple circuit for the precise measurement of resistors known as the Wheatstone Bridge. The circuit, invented by Samuel Hunter Christie (1784-1865) in 1833, was named after Sir Charles Wheatstone (1802-1875) who ‘found’ and popularized the arrangement in 1843. It consists of an electrical source and a galvanometer that connects two parallel branches, containing four resistors, three of which are known. One parallel branch consists of a known and unknown resistor (R4), while the other branch contains two known resistors.
10. 10. • Kirchoff ’s Current Law yields: I0 - I1 - I2 = 0 I1 - I5 - I3 = 0 I2 + I5 - I4 = 0 I3 + I4 - I0 = 0 • And Kirchoff ’s Voltage Law yields: I2R2 - I5R5 - I1R1 = 0 I5R5 + I4R4 - I3R3= 0 I2R2 + I4R4 - E = 0 I1R1 + I3R3 - E = 0
11. 11. In this case, we observe a circuit that has a 5-volt power supply with different loops, and its resistors. Notice now that we have three loops drawn, all rotating clockwise. Next, we must drawn loops in which the current in the circuit travels, called I1, I2, and I3. I1, I2, and I3 are all current loops (measured in Amps).
12. 12. We start with the general equation, 𝑛=1 𝑛 𝐼𝑛 ∗ 𝑅𝑛 = 𝑉 Where V is the voltage, I is the current around a loop, and Rn is the total resistance of the path for the given current In. Next, we want to look at each loop, and set up an equation, which uses all paths that touch the loop multiplied by their total resistances where they touch that path. Observe the following equations: 18I1 – 2I2 -5I3 = 5 -2I1 + 5I2 -3I3 = 0 -3I1 – 5I2 +9I3 = 0 The coefficients for I1, I2, and I3 are all the total resistances for those loops, which have unknown current, and they are set equal to the total potential difference (voltage) around that loop. We can then put these equations into an augmented
13. 13. When we put the system is put into an augmented matrix, we get the following: 18 − 2 − 5 5 −2 5 − 3 0 −5 − 3 9 0 When we row reduce this matrix, we get 1 0 0 0.4215 0 1 0 0.3864 0 0 1 0.3630 From this, we can determine what the current through I1, I2, and I3 are.
14. 14. Thank You