SOME MATHEMATICAL MODELS IN CIRCUIT THEORY
BY
NURT9JA
A PROJECT SUBMITTED TO THE DEPARTMENT OF MATHEMATICS, -------------------
UNIVERSITY, NIGERIA IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF
THE DEGREE OF BACHELOR OF SCIENCE (B Sc. Hons.) IN MATHEMATICS.
SEPTEMBER, 2016
DECLARATION
…………………
…………………..
Signature. Date.
CERTIFICATION
This project
entitled “Some Mathematical Models in
Circuit Theory” by NURT9JA meets the requirements governing the award of
the degree of Bachelor of Science in Mathematics and is approved for its
contribution to knowledge and literary presentation.
........................................................ ……………
Dr. -----------
Date
Supervisor
…………………………………
…………….
Prof. ----------
Date
Head of
Department
External Examiner
Name:…………………………………………….. ……………
Signature:………………………………………… Date
DEDICATION
This
project is dedicated to
ACKNOWLEDGMENT
ABSTRACT
This project deals
with the derivation and analysis of first order ordinary differential equations
in RC and RL circuits and second order ordinary differential equations
in RLC circuits. In my
project, series circuits
are associated with
voltage sources and
parallel circuits are
associated with current
sources. In chapter
four which involves
the derivation and
analysis aspect of
the project, the
order is:
DERIVATION: Derivation
of first order
ordinary differential equations
in RC circuits,
derivation of first
order ordinary differential
equations in RL
circuits and then
derivation of second
order ordinary differential
equations in RLC
circuits.
ANALYSIS: Analysis
of derived first
order ordinary differential
equations in RC
circuits, analysis of
derived first order
ordinary differential equations
in RL circuits
and then analysis
of derived second
order ordinary differential
equations in RLC
circuits.
Analysis of
all first order
ordinary differential equations
and all second
order ordinary differential
equations in my project can
be achieved using
only basic integration
methods and basic
algebra methods.
Contents
Declaration…………………………………………………………………i
Certification……………………………………………………………….ii
Dedication……………………………………………………………....iii
Acknowledgement………………………………………………………iv
Abstract…………………………………………………………………vii
Chapter1
General
Introduction…..............................................................................................1
1.1Introduction………………………………………….
1
1.2 Aim And Objective …………………………………..1
1.3 Significance of The study……………………………1
1.4
Scope And
Limitation ………………………………1
1.5
Basic Definitions
…………………….………………2
1.5.1 Differential
Equations ……………..………………2
1.5.2 Circuit Theory ............................................................4
1.6
Example…………………………………………………6
Chapter 2
Literature Review……………………………………………9
Chapter 3
Methodology and Tools/Data
presentation……………...10
Chapter 4
Analysis
and Discussion of
results……………....................11
4.1 Derivation of
equations involving voltage Sources .....12
4.2 Derivation of
equations involving current
sources……13
4.3 Analysis of
equations involving voltage sources………16
4.4 Analysis of
equations involving current
sources…….17
4.5 Discussion of
results………………………………..….18
Chapter 5 Summary,
Conclusion And Recommendation…………….19
5.1Summary…………………………………………………….19
5.2Conclusion………………………………………………..…19
5.3Recommendation……………………………………………19
REFERENCES………………………………………………………20
APPENDICES……………………………………………………….21
CHAPTER 1: GENERAL INTRODUCTION.
1.1.Introduction
A model is a description of a phenomenon. A differential equation
is an example of a mathematical model. Differential equations have many
applications in the biological and physical sciences.
1.2 Aim and Objective
The aim of the project work is to derive and analyze differential
equations associated with series and parallel arrangements of RC, RL and RLC
circuit.
1.3 Significance of the study
RC, RL and RLC circuit are simple but very significant in circuit
theory. This project work will provide the Electrical Engineer with an easy
reference.
1.4 Scope and Limitation
Though RC, RL and RLC circuits are important, there are many other
circuits in circuit theory. This project work deals only with the RC, RL and
RLC circuits.
1.5 Basic definitions
1.5.1 Differential equations
·
Differential
equation: A differential equation is an equation in terms of a derivative that models a phenomenon.
·
Ordinary
differential equation: This is a differential equation involving a function of
one variable.
·
Partial
differential equation: This is a differential equation involving a function of
more than one variable.
·
Order: This is
the number of times required to differentiate a function to get a derivative.
·
Solution: This
is a function which satisfies an equation.
·
Digits: The ten numbers: 0,1,2,3,4,5,6,7,8,9 are
called digits.
·
Decimal Places:
The number of
digits after the
decimal point is
the number of
decimal places.
·
Significant Figures: The
number of digits
in a calculation
is called the
number of significant
figures.
·
Mathematical Operations
on numbers: The 6 operations
on numbers are
addition (increase in number),
subtraction (decrease in number),
multiplication (repeated addition)
and division.
·
Arithmetic: Arithmetic
deals with operations
on numbers.
·
Algebra: Algebra
involves manipulating numbers
using placeholders for
the numbers.
·
Trigonometry: When
there is rotation
about a point,
there is always
a change in
the angle at
the point. Scenarios
that involve rotations
described as well
as related issues
are studied under
trigonometry.
·
Expression: A
mathematical expression is a mathematical
statement involving numbers
and mathematical operations.
·
Equation: An
equation is a way of
saying two expressions
are the same
by putting one
on the left
of an equality
sign (=) and
the other on
the right of
the equality sign.
·
Natural numbers:
Natural numbers are
counting numbers.
·
Number System:
A number system
is a way
of counting developed
at a particular
part of the
World.
·
Positive Number:
A positive number
is a number
in a number
system that is
greater than a
number in the
number system called
the identity.
·
Identity: The
identity in a
number system is
a neutral number
in the number
system that is
neither positive nor
negative.
·
Inverse: The
inverse of a
number in a
number system is
another number in
the number system
which when added
or multiplied to
it yields the
identity in the
number system.
·
Additive Inverse: The
additive inverse of a number
in a number
system is a
number in the
number system which
when added to
it yields the
identity in the
number system.
·
Multiplicative Inverse:
The multiplicative inverse
of a number
in a number
system is a
number in the
number system which
when multiplied by
it yields the
identity in the
number system.
·
Set: A
set is a
group.
·
Quantity: A
quantity is a
number.
·
Variable: A
variable is a
placeholder for one
or more numbers.
·
Independent Variable:
A variable may be independent
of all quantities
in the situation
under consideration. In
such a case,
it is called
the Independent Variable.
·
Dependent Variable:
A variable may
also rely on
the Independent Variable.
In such a
case, it is
called a dependent
variable.
·
Function: A
function is a
relationship between one
number in a
set and another
number in another
set. The relationship
may be expressed
verbally, in the
form of an
equation, in the
form of a
table involving a
column for different
values of the
Independent Variable and
corresponding values for
the dependent variable,
or in the
form of a
graph for the
table of values.
Circuit theory
·
Voltage:
Voltage is the
work done per
unit charge. Voltage is
measured in volts
(V).
·
Current:
Current is the speed of flow of charge.
Current is measured
in amperes (A).
·
Field: A
field is an
effect on an
object within a
specified radius.
·
Electricity: Electricity
is a flow
of electrons under
the influence of
a field.
·
Charge: Charge
refers to the
amount of electricity
carried by electrons.
It is measured
in Coulombs (C).
·
Resistance:
Resistance is the voltage per unit current.
Resistance is usually
denoted by R. A device
with resistance is
called a resistor.
The unit of
resistance is the
ohm (
.
·
Capacitance:
Capacitance is the current per unit rate of change of voltage. Capacitance
is usually denoted
by C. A
device with capacitance
is called a
capacitor. The unit
of capacitance is
the farad (F).
·
Inductance:
Inductance is the voltage per unit rate of change of current. Inductance
is usually denoted
by I. A
device with inductance
is called an
inductor. The unit
of inductance is
the henry (H).
·
Impedance: Impedance is a
general term for
resistance, capacitance and
inductance.
·
Series
Arrangement: Two or
more electrical components
are connected in
series if the
same current flows
through them.
·
Parallel
Arrangement: Two or
more electrical components
are connected in
parallel if they
are connected between
the same terminals
and they have
the same voltage
across them.
·
Direct Current:
Current is direct
if it moves in only
one direction.
·
Alternating Current:
Current is alternating
if it moves
in two directions:
forward and backward.
·
Alternating Current
Source: A source
of electrical energy
is an alternating
current source if
it gets it’s
stored energy from
another source that
supplies it with
kinetic energy in
the form of
an alternating current
of electrons.
·
Principle of
operation of Kainji
Dam: The top
of Kainji Dam
is very far
from the bottom.
Because of this
fact, there is
an enormous amount
of potential energy
at the top
of Kainji Dam
which is gradually
converted to kinetic
energy as water
falls from the
top of Kainji
Dam to the
bottom. There will
be other energy
conversions as water
falls from the
top of Kainji
Dam to the
bottom some of
which may be
energy losses in
the form of
heat energy. This
enormous amount of
energy is converted
by a very
sophisticated electronic device
to electrical kinetic
energy in the
form of alternating
current of electrons
at very high
voltage which is
transmitted via wires
to our homes.
There are transformers
at strategic locations
that bring down
the voltages of
the alternating current
of electrons to
safe levels before
the alternating currents
reach our homes.
·
Direct Current
Source or Battery: A
source of electrical
energy is called
a direct current
source or battery
if it acquires
stored energy from
another source via
a current of
electrons that moves
in only one
direction.
·
Voltage source:
A voltage source,
with a voltage
of say 5
volts ( 5
joules per coulomb),
is a source
of electrical energy
that supplies some
coulombs of freely
moving electrons that
dissipate an energy
of 5 joules
for each coulomb
of electrons. This
energy is distributed
amongst the electrical
components connected to the voltage
source via wires
or otherwise according
to laws in
circuit theory.
·
Current source:
A current source,
with a current
of say 5 amperes (
5 coulomb per
second ), is
a source of
electrical energy that
supplies 5 coulombs
of electrons after
every second.
1.5
Example.
Example with Direct
Current Source:
Suppose there is
a 10 ohm
resistance in series
with a 5
Farad capacitance. Both
get their energy
from a 10
volt direct current
source. Let us
find the equivalent
impedance of the
resistor and capacitor
that will receive
the energy and
the energy received
by the resistor
and capacitor. I
will also include
in the solution
some related analysis.
Solution:
Wires connect all
the electrical components
with the direct
current source in
series with a
resistor and capacitor. Let me
denote the resistance
by R, the capacitance
by C and
the energy by W.
EQUIVALENT IMPEDANCE:
The equivalent impedance
of the resistor
and capacitor is
given by
(R x C)/(R+C) = (10x5)/(10+5)
= 50/15 =
(25+25)/(15) = (25/15)+(25/15) =
(25/15)x2
= (5/3)x2 =
10/3 = 3.33 (2
Decimal places)
ENERGY SUPPLIED BY
THE DIRECT CURRENT
SOURCE: The energy
supplied by the
direct current source
to the resistor
and capacitor is
given by: W=V
x I x
t where V
is the equivalent
voltage of the
resistor and capacitor
and also the
voltage of the
source, I is
the current flowing
through both the
resistor and capacitor
in series, t
is the time
within which the
current flow through
the resistor and
capacitor took place.
The voltage is
supplied by the
direct current source
and the supplied
voltage is distributed
between the resistor
and capacitor. The
current flowing through
the capacitor is:
I2 = C
where
is
the speed at
which the voltage
of the direct
current source is changing.
Whenever
there is a
flow of electrons
in a circuit,
there is voltage
and there is
current. The voltage
may be constant
or changing and
the current may
also be constant
or changing.
The
fact that the
direct current source
has a voltage
of 10 Volts
(10 joules per
coulomb) means that
a group of
electrons with a
charge of 1
coulomb flowing from
the direct current
source dissipate an
energy of 10
joules when moving
through 1 meter
under the influence
of a force
of 10 newtons.
The distance covered
is 1 meter
and the force
is 10 newtons
because 10 joules
= 10 newtons
x 1 meter.
Since
the voltage of
the direct current
source is constant
at 10 volts
and the electrons
flowing do not
change direction, this
means that there
is no change
in voltage.
Because
there is no
change in voltage,
Thus,
there is no
current flowing through
the capacitor and
all the current
flows through the
resistor.
Suppose
the current supplied
by the direct
current source is 10 amperes
and it flows
for 10 seconds,
then, the energy
supplied by the
direct current source
is: W =
V x I
x t =
10 x 10
x 10 =
1000 joules.
CHAPTER 2: LITERATURE REVIEW
Kirchhoff’s voltage
law and Kirchhoff’s current law
have been used. Concepts from THE PRINCIPLES OF NATURAL PHILOSOPHY have also been used. I referred to the
works by Murray Spiegel; Paul Blanchard, Robert L. Devaney and Glen R.
Hall; Raymond A.
Barnett, Michael R.
Ziegler and Karl
E. Byleen; Jerry
J. Anderson and
others.
CHAPTER 3:
METHODOLOGY AND
TOOLS/DATA
PRESENTATION
The equations were analyzed using
integration and algebra
methods.
CHAPTER 4
ANALYSIS AND
DISCUSSION OF RESULTS
The
voltages and currents
are functions of
time. R,
C
and L are
parameters.
4.1 Derivation of
equations involving voltage sources; series circuits
a)
RC Circuit
The RC circuit consists of a resistor, a
capacitor and a
voltage source.
b) RL Circuit
The RL circuit consists of a resistor, an inductor
and a
voltage source.
c)
RLC circuit
The
RLC circuit consists of a resistor, an inductor, a capacitor and a voltage
source.
4.2
Derivation of equations
involving current sources; parallel circuits
a)
RC Circuit
The RC circuit
consists of a resistor, a capacitor and a current source.
b)
RL Circuit
The RL circuit
consists of a resistor, an inductor and a current source.
c)
RLC Circuit
The RLC circuit consists of a resistor, an inductor, a capacitor
and a current source.
4.3
Analysis of equations involving voltage sources;
series circuits
a)
RC Circuit
a)
RL Circuit
b)
RLC Circuit
4.4 Analysis of
equations involving current sources;
parallel circuits
a)
RC Circuit
b)
RL Circuit
c)
RLC Circuit
4.5 Discussion
of results
From
the derivations and
analysis above, for
the cases considered
it is evident
that once a
circuit has been
analyzed using Kirchhoff’s
laws or another (other) method(s),
the solution can
be found using
simple or multiple
integration.
CHAPTER 5: SUMMARY, CONCLUSION AND
RECOMMENDATION.
5.1 Summary: The project work is concerned with information on
important circuits in circuit theory.
5.2 Conclusion: The most difficult concepts are simple to analyze.
Kirchhoff’s current law and Kirchhoff’s voltage law are simple to understand
but very powerful. For example, the two laws were very helpful in the
derivation part of the project work.
5.3 Recommendation: I recommend that other simple circuits in
circuit theory should be analyzed by other students.
REFERENCES
Cited Literature
1.
Spiegel, M.,
(1987) ADVANCED MATHEMATICS FOR
ENGINEERS & SCIENTISTS, SI (metric) ed., Schaum’s Outline Series,
McGraw- Hill, Singapore.
2.
Blanchard, P.,
et al., (2002) Differential
Equations, 2nd ed., Thomson Learning.
3.
Barnett, R.A.,
et al., (2008)
Precalculus, 6th ed., McGraw-Hill International
Edition, United States
of America.
4.
Anderson, J.J.,
et al., (1979)
The World Book
Student Handbook, Student Guide, World Book-Childcraft International, Inc.
A subsidiary of
The Scott &
Fetzer Company, United
States of America.
5.
Hornby, A.S.,
(2006) Oxford Advanced
Learner’s Dictionary of
Current English, OXFORD
UNIVERSITY PRESS, Oxford,
United Kingdom.
APPENDIX
SQ3R STUDY METHOD
The SQ3R method
is one of
the most effective
and efficient methods
of study. The
‘S’ stands for
survey, the ‘Q’
stands for question,
the ‘3R’ means ‘R,
R, and R’ and stands
for read, recite
and review. I
shall give an
overview of the
method.
SURVEY:
The first step
is to thumb
through the book,
chapter or material.
QUESTION:
The second step
is to convert
each heading of the chapter
or material into
a question. You
do not have
to put down
the questions.
READ:
The third step
is to read
the chapter or
material.
RECITE:
The fourth step
is to close
the book or
material and put
down all you
can remember about
what you read.
REVIEW:
The final step
is to revise
the chapter or
material.
DISTINCTION BETWEEN A
SET AND A
SEQUENCE
A set is
a collection and
a sequence is
a set of
numbers whose members
are determined by
a formula. The
dependent variable in
the formula for
a sequence is
the position of
the member of
that sequence. A
set is represented
by curly brackets
and a sequence
is represented by
braces.
Strategy for solving scientific
problems
·
I shall give
a method I
believe is effective
in tackling scientific
questions.
·
The first step
is to thoroughly
read and understand
the problem in
good time.
·
The second step
is to define
all quantities including
the independent variable
which is usually
time, the dependent
variables and the
parameters.
·
The third step
is to formulate
or model the
problem using prior
knowledge.
·
The final step
is to use
techniques developed by
prior scientists (like
Isaac Newton and
Albert Einstein) to
tackle the problem.
A Concise Discussion
on Space
A space is
simply a set
of numbers together
with a function
defined on the
set of numbers.
Symbols are used
to denote the
set of numbers
and the function
defined on the
set of numbers.
A space is
usually represented as an ordered
pair where one
of the ordered
pairs is the
symbol representing the
set of numbers
on the left
and the other
of the pairs
is the symbol
representing the function
defined on the
set of numbers
on the right.
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