Application of Linear Algebra in Electrical Circuit
- 1. APPLICATION OF LINEAR ALGEBRA IN ELECTRICAL CIRCUIT Welcome
- 2. PRESENTED TO Md. Mosfiqur Rahman Senior Lecture in Mathematics Department of GED Daffodil International University Presented by Gazi Md Badruzzaman JHON Electronic & Telecommunication Engineering ID:171-19-1937 Daffodil International University
- 3. CONTENTS……! ☼Introduction ☼Linear Algebra & various fields ☼History of Linear Algebra ☼Electrical Circuits ☼Electrical circuit In Linear Algebra ☼Gaussian Elimination ☼ The Wheatstone Bridge
- 4. INTRODUCTION • This presentation is mainly about to let us all know that how electrical circuits works on applications of LINEAR ALGEBRA • All you need to be a inventor is a good imagination and a pile of junk. by:- THOMAS EDISON
- 5. 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.
- 6. 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 Applications of Linear Algebra in various fields.
- 7. LINEAR ALGEBRA History: The study of linear algebra first emerged from the study of determinants, Determinants were used by Leibniz in 1693 Gabriel Cramer devised Cramer's Rule for solving linear systems in 1750. Gauss further developed the theory of solving linear systems by using Gaussian elimination In 1844 Hermann Grassmann publish "Theory of Extension“ which founded on linear algebra In 1848, James Joseph Sylvester introduced the term matrix Linear algebra first appeared in American graduate textbooks in the 1940s and in undergraduate textbooks in the 1950s
- 8. ELECTRICAL CIRCUITS † Electrical circuit is nothing but just a combination of transistor, capacitor, diodes, etc. including some logic gates. † Each component has it’s own specification. † And through which we get to know what currents and voltages are. † An electrical circuit is a path in which electrons from a voltage or current source flow.
- 10. LINEAR ALGEBRA IN ELECTRICAL CIRCUITS • 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.
- 11. GO ON… • 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.
- 12. 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.
- 13. GO ON… This method is known as Gaussian Elimination. Now, for large circuits, this will still be a long process to row reduce to echelon form. 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
- 14. 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.
- 15. • 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
- 16. 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).
- 17. n ∑ In *Rn=V n=1 We start with the general equation 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 matrix and put the matrix into rref
- 18. 18 -2 -5 5 -2 5 -3 0 -5 -3 9 0 1 0 0 0.4215 0 1 0 0.3864 0 0 1 0.3630 When we put the system is put into an augmented matrix, we get the following: When we row reduce this matrix, we get From this, we can determine what the current through I1, I2, and I3 are.