Course Material
OF
FUNDAMENTALS OF ELECTRICAL
ENGINEERING
BEE-101/BEE-201
Session 2022-23
Course Coordinator:
Mr. Vinod Kumar
Course Teachers:
Mr. Bijendra Kumar
Dr. V.S. Gupta
Dr. Sunil Kumar Chaudhary
Mr. Manish Shrivastava
Mr. Ravindra Kumar Yadav
Dr. Bhuvnesh
BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida
Unit 1: DC Circuits
CO1: Apply Kirchhoff’s laws and network theorems in solving DC circuits
Contents: Electrical circuit elements (R, L and C), Concept of active and passive elements, voltage
and current sources, concept of linearity and linear network, unilateral and bilateral elements,
Kirchhoff‟s laws, Loop and nodal methods of analysis.
1. Electrical circuit elements (R, L and C): The interconnection of various electric
elements in a prescribed manner comprises as an electric circuit in order to perform a
desired function. The electric elements include controlled and uncontrolled source of
energy, resistors, capacitors, inductors, etc. Analysis of electric circuits refers to
computations required to determine the unknown quantities such as voltage, current and
power associated with one or more elements in the circuit. To contribute to the solution
of engineering problems one must acquire the basic knowledge of electric circuit analysis
and laws. We shall discuss briefly some of the basic circuit elements and the laws that
will help us to develop the background of subject.
a) Resistor: Resistor is a dissipative element, which converts electrical energy into heat when
the current flows through it in any direction. The law governing the current into and voltage
across a resistor is:
(i)
The relationship is known as Ohm’s law.
But resistor can be regarded as linear only within the specified limits, outside which
the behavior becomes non-linear. The resistance of a resistor is temperature
dependent and rises with temperature.
Mathematically it can be represented as:
(ii)
Where = Resistance at and = Resistance at
Temperature coefficient and it may be positive and negative both
Temperature in
And power dissipated by resistor is
Watts
Resistor is represented by the symbol
Unit of Resistance is ohm (𝛀)
b) Capacitor (C): It is a two terminal element that has the capability of energy storage in
electric field. The law governing the relationship of capacitor is:
(iii)
After integrating equation (iii), we get
BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida
∫ (iv)
Where Capacitor voltage at , for initially uncharged capacitor
Hence, ∫ (v)
The above expressions show that the voltage of a capacitor cannot change
instantaneously.
Energy stored in capacitor can be represented by
∫ ∫ ∫ (ix)
Capacitor is represented by the symbol
Unit of Capacitance is Farad (F)
c) Inductor (L): It is a two-terminal storage element in which energy is stored in the
magnetic field. The relation of an inductance is:
(vi)
After integrating expression (vi), we get
∫ (vii)
Where Inductor current at , for initially if current through inductor
Hence, ∫ (viii)
The above expressions show that the current through an inductor cannot change
instantaneously.
Energy stored in inductor can be represented by
∫ ∫ ∫ (ix)
Inductor is represented by the symbol
Unit of Inductance is Henry (H)
2. Concept of active and passive elements:
Electrical Network: Any possible combination of various electric elements
(Resistor, Inductor, Capacitor, Voltage source, Current source) connected in any
manner what so ever is called an electrical network. We may classify circuit
elements in two categories, passive and active elements.
BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida
Electrical Circuit: An electric circuit is a closed energized electric network. It
means circuit must have closed path with energy sources. From the above example,
we can say that fig 1 and fig 2 are electric networks but only fig 2 is electric circuit.
It means, electric circuit is always an electric network but electric network may or may
not be an electric circuit.
Passive Element: The element which receives energy (or absorbs energy) and then
either converts it into heat (R) or stored it in an electric (C) or magnetic (L ) field is
called passive element, and the network containing these elements without energy
sources are known as passive network. Examples are resistor, inductor, capacitor,
transformer etc.
Active Element: The elements that supply energy to the circuit is called active
element and the network containing these sources together with other circuit
elements are known as active network. Examples of active elements include voltage
and current sources, generators, and electronic devices that require power supplies.
A transistor is an active circuit element, meaning that it can amplify power of a
signal.
3. Energy Sources (Voltage and Current Sources): There are two types of energy
sources namely Voltage Sources and Current Sources.
Energy Sources
Voltage Source Current Source
Independent Dependent Independent Dependent
Voltage Source Voltage Source Current Source Current Source
Voltage Dependent Current Dependent Voltage Dependent Current Dependent
Voltage Source Voltage Source Current Source Current Source
(VDVS) (CDVS) (VDCS) (CDCS)
BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida
Here, we shall study only about independent voltage source and independent current
source.
a) Independent Voltage Source: A hypothetical generator which maintains its value
of voltage independent of the output current. It can be represented as:
Fig: Ideal DC Voltage Source Fig: Practical DC Voltage Source
If the value of internal resistance will be zero, then the voltage source is called as
ideal voltage source. The V-I characteristics for ideal and practical voltage source is
given below:
Fig: V-I Characteristic of Voltage Source
b) Independent Current Source: A generator which maintains its output current
independent of the voltage across its terminals. It can be represented as:
Fig: Ideal DC Current Source Fig: Practical DC Current Source
if the value of internal resistance will be infinity, then the current source is called as
ideal current source. The V-I characteristics for ideal and practical current source is
given below:
BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida
Fig: V-I Characteristic of Current Source
4. Concept of Linearity and Linear Network: For a network to be linear, it should
have to follow the principle of superposition and homogeneity both.
Principle of Superposition: An element or circuit obeys the principle of
superposition if the net effect of the sum of causes equals the sum of their individual
effects.
Mathematically, let cause x and effect y be related as:
(i)
Let the cause be scaled by a factor . Then the functional relationship obeys
homogeneity, if
(ii)
Consider two causes , then
Let the combined effect of these two causes be scaled by and respectively. The
principle of superposition then yields if:
(iii)
If homogeneity is also satisfied, then
(iv)
A functional relationship is said to be linear if it obeys both superposition and
homogeneity. Any element governed by such a functional relationship is linear. A
circuit composed of such elements would also be linear.
5. Unilateral and Bilateral Elements:
Bilateral Elements: If by reversing the terminal connections of an element in a
circuit, the circuit response remains same. Such elements are known as bilateral
elements. Examples are Resistor, Inductor, Capacitor etc.
Unilateral Elements: If by reversing the terminal connections of an element in a
circuit, the circuit response gets change. Such elements are called as unilateral
elements. Examples are Voltage Source, Current Source, Diode etc.
BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida
6. Kirchhoff‟s laws: There are two types of Kirchhoff’s Law.
1. Kirchhoff’s First Law or Kirchhoff’s Current Law (KCL)
2. Kirchhoff’s Second Law or Kirchhoff’s Voltage Law (KVL)
1. Kirchhoff’s First Law or Kirchhoff’s Current Law (KCL): Kirchhoff’s current
law states that, in a given electric circuit, algebraic sum of all the currents meeting at
a junction is always zero. In another way we can say that, the total current flowing
towards a junction is equal to the total current flowing away from that junction. This
law works on the principle of conservation of charge.
Sign Convention: If we take direction of current towards the junction as positive (+)
sign then direction of current away from the junction will be taken as negative (-)sign
or vice-versa.
According to Kirchhoff’s Current Law in the above circuit diagram:
2. Kirchhoff’s Second Law or Kirchhoff’s Voltage Law (KVL): Kirchhoff’s voltage
law states that, “ In any electric circuit, the algebraic sum of the voltage drops across
the circuit elements of any closed path (or loop or mesh) is equal to the algebraic
sum of the EMFs in the path”.
BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida
In other words, “ The algebraic sum of all the branch voltages around any closed
path or closed loop is always zero”. This law works on the principle of conservation
of energy. Limitation of this law is that it can only be applied to planner network.
Sign Convention: If we take voltage rise with positive (+) sign then voltage drop will
be taken with negative (-) sign or vice-versa. When current direction will be from
negative terminal to positive terminal, voltage will rise and vice-versa. In all the
passive elements current entering terminal is taken as positive and current leaving
terminal is taken as negative.
According to KVL in the above circuit diagram:
7. Loop and Nodal Methods of Analysis: For study of Loop and Nodal Methods of
Analysis, knowledge of basic fundamentals are essential.
Some Basic Definitions:
1. Node: A node of a network is an equipotential surface at which two or more
circuit elements are joined.
2. Junction: A junction is that point in an electric circuit where three or more
elements are joined.
So, we can say that junction is always a node but node may or may not be a
junction.
3. Loop: A loop is any closed path of the electric network.
4. Mesh: A mesh is the most elementary form of loop, and it cannot be further
subdivided into other loops.
So, we can say that mesh is always a loop but loop may or may not be a mesh.
BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida
5. Lumped Network: A network in which physically separate resistors, capacitors
and inductors can be represented.
6. Distributed Network: One in which resistors, capacitors, and inductors cannot
be physically separated and individually isolated as separate elements. For example,
Transmission Line.
Loop or Mesh Analysis Method: Mesh analysis is also known as loop analysis
method. Mesh analysis is used to find the currents and voltages in a particular
circuit.
Suppose in a particular electrical circuit , total number of branches are b, total
number of nodes are n and total number of junctions are j, then total number of
meshes ‘m’ can be calculated by using the following expression:
(i)
Also, (ii)
Note: One thing make sure, when we consider nodes in the given electric circuit
then branches will be counted according to the number of nodes, and equation (i) is
used to calculate the total number of meshes. When we consider junction in that
electrical circuit then branches will be counted according to junctions, and equation
(ii) is used to calculate the total number of meshes. Using both the methods same
number of meshes will be found for a particular circuit.
The independent mesh equations can be obtained by applying KVL to each
independent mesh.
Mesh current is that current which flows around the perimeter of a mesh. Mesh
currents may or may not have a direct identification with branch currents.
Mesh currents on the other hand, are fictitious quantity which are introduced
because they allow us to solve problems in terms of a minimum number of
unknowns.
Procedure of Mesh Analysis Method:
Step 1: Draw the circuit in which mesh currents or branch currents have to find.
Calculate number of independent mesh equations by using the formula given below:
(i)
Also, (ii)
BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida
Step 2: Assume independent mesh currents for each mesh. You can choose any
direction of the mesh current, but once you have chosen the direction of current, it
should remain same throughout the question.
Step 3: Apply KVL for each mesh and write the expressions in terms of unknown
mesh currents.
Step 4: Now, solve the equations by any method, either by simultaneous equation
method or Cramer’s Rule and find the unknown values.
Example 1: Find the current through 'ab-branch' ( ) and voltage ( ) across the
current source using Mesh-current method in the given circuit diagram.
BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida
BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida
Nodal Methods of Analysis: Circuit analysis by this methods are solved by using
the KCL at the junction of a particular given circuit.
Suppose, total number of junctions are j in a particular electrical circuit, then total
number of node equations N can be calculated by using the following formula:
Total number of Node Equations (i)
Procedure of Nodal Methods of Analysis:
Step1: Draw the electrical circuit in which node voltage or branch currents has to
find using this method and calculate the total number of Node equations using the
formula: (i)
Step 2: Assume independent node voltages for each junction except reference,
because at reference junction voltage will always be zero.
Step 3: Apply KCL for each junction except reference.
Step 4: Finally solve the equations using different methods.
Example 2: Find the value of the current I flowing through the battery using ‘Node
voltage’ method in the given circuit.
BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida
BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida
