application of linear algebra in electrical circuit
TRANSCRIPT
APPLICATION OF LINEAR ALGEBRA IN ELECTRICAL
CIRCUIT
Welcome
PRESENTED TO
Md. Mosfiqur RahmanSenior Lecture in Mathematics Department of GEDDaffodil International University
Presented by
Gazi Md Badruzzaman JHONElectronic & Telecommunication Engineering ID:171-19-1937Daffodil International University
CONTENTS……!
☼Introduction
☼Linear Algebra & various fields
☼History of Linear Algebra
☼Electrical Circuits
☼Electrical circuit In Linear Algebra
☼Gaussian Elimination
☼ The Wheatstone Bridge
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
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.
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.
LINEAR ALGEBRAHistory:
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
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.
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.
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.
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.
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
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.
• 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
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).
n
∑ In *Rn=Vn=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 -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.