SSDN
FUNDAMENTALS OF ELECTRICAL
ENGINEERING
(BEE 201)
SESSION – April 2025 – July 2025
UNIT - 1
Dr. Pavan 1
Khetrapal
Contact
UNIT -1: DC CIRCUITS Hours
Electrical circuit elements (R, L and C),
Concept of active and passive elements,
voltage and current sources, concept of
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linearity, unilateral and bilateral elements.
Kirchhoff’s laws, Mesh and nodal methods of
analysis.
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ENERGY SOURCES
An energy source in an electrical circuit is a device or system that supplies
electrical energy.
It establishes and maintains an electric potential difference (voltage)
or current flow within the circuit.
ENERGY SOURCES
Classification by
Classification by Classification by
DEPENDENC
IDEALITY OUTPUT TYPE
Y
DEPENDENT
INDEPENDEN
IDEAL PRACTICAL (CONTROLLED VOLTAGE CURRENT
T
SOURCES (NON-IDEAL) ) SOURCE SOURCE
SOURCES
SOURCES SOURCES
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VOLTAGE AND CURRENT SOURCES
An Energy Source or simply a source can be IDEAL (i.e. Theoretical) or it
can be PRACTICAL.
IDEAL VOLTAGE SOURCE PRACTICAL VOLTAGE SOURCE
• An Ideal Voltage Source is a • A Practical Voltage Source (also
theoretical voltage source that called a real voltage source) is a real-
maintains a constant voltage across world device that approximates an ideal
its terminals, regardless of the voltage source, but has a small
current drawn from it, and internal resistance.
• The output voltage drops slightly
• It has zero internal resistance.
when current is drawn due to the internal
resistance.
• Example: A 12 V battery with 1 Ω internal
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resistance
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IDEAL CURRENT SOURCE PRACTICAL CURRENT SOURCE
IDEAL CURRENT
SOURCE
IL
V – I Characteristic
IS
PRACTICAL
CURRENT SOURCE
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Ideal voltage and current sources can be further described as
either independent sources or dependent sources.
An Independent Voltage Source establishes a voltage in a circuit
without relying on voltages or currents elsewhere in the
circuit. The value of the voltage supplied is specified by the value of
the independent source alone.
An Independent Current Source establishes a current in a circuit
without relying on voltages or currents elsewhere in the
circuit. The value of the current supplied is specified by the value of
the independent source alone.
In contrast, a Dependent Source establishes a voltage or current
whose value depends on the value of a voltage or current elsewhere
in the circuit. Shown by Diamond shaped symbol.
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TYPES OF DEPENDENT SOURCES
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Contd.
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EXAMPLES
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MESH AND NODAL ANALYSIS
A LOOP is a closed conducting path through a circuit in which no element is
encountered more than once. [Starting at an arbitrarily selected point, we trace
a closed path in a circuit through selected basic circuit elements and return to
the original point without passing through any intermediate node more than
once.]
Choosing point a as the starting point
and tracing the circuit clockwise, we
form the closed path by moving
through points d, c, b, and back to
point a.
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a b c
a b c
h d
f e d
f
g e
LOOP 1 a-b-e-f-a
LOOP 2 b-c-d-e-b
LOOP 3 a-b-c-d-e-f-a Total Number of Loops -
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Total Number of Loops -
3
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MESH AND NODAL ANALYSIS
A MESH is a loop that does not contain any other loop within it. [A MESH is a
closed electrical pathway that does not contain other closed physical pathways]
Total Number of Meshes
-1 Total Number of Meshes
-4
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Total Number of Meshes
-2 Total Number of Meshes
-3
14
Total Number of
Meshes - 3
Total Number of Meshes
-3
Total Number of Meshes
-2 Total Number of Meshes - 2 15
A SUPERMESH results when two meshes have a (dependent or
independent) current source in common.
SUPERMES
H
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MESH ANALYSIS
Mesh Analysis is used to find mainly the mesh currents
and other variables if required.
A mesh current is defined as a current that flows only around
the perimeter of a mesh.[A mesh current is a hypothetical
current that flows around a mesh in an electric circuit].
Mesh currents are not always the actual branch currents
but are used to calculate them.
KVL is implemented to find the mesh currents.
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SOLVING SIMULTANEOUS EQUATIONS
Ques Ques
: :
9 𝑖1 −3 𝑖2 =42
− 3 𝑖 1+7 𝑖2=10
Answer:
Ques
:6 𝑖− 2 𝑖2= 7
1 Answer:
−2 𝑖1 +7 𝑖 2=−3
Answer:
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Ques: Write down the equations for mesh
current using KVL.
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Ques: Find the mesh currents
i1 and i2 and branch
currents I1, I2, I3 for the
circuit shown.
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Ques: Find the mesh currents and branch currents
Equations (1) and (2) are two simultaneous
equations and can be solved either by substitution
method or determinant method. On solving
equations (1) and (2), the mesh currents are
obtained as
BRANCH CURRENTS:
• The current flowing through 6 Ω resistor is
6A
• The current flowing through 4 Ω resistor is
4A 21
• The current flowing through 3 Ω resistor is
Ques: Find the mesh
currents for the circuit
shown.
Equations (1), (2) and are three simultaneous equations and can
be solved either by substitution method or Cramer’s rule. On
solving equations (1) , (2) and , the mesh currents are obtained as
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Since the dependent source is controlled
by the unknown voltage vx , we are faced
with two equations in three unknowns.
The way out of our dilemma is to
construct an equation for vx in terms of
mesh currents, such as
[ 𝟑]
We simplify our system of equations by
substituting Eq. [3] into Eq. [1], resulting in
[ 𝟏]
𝟒 𝒊 𝟏=𝟓 [ 𝟒]
Solving, we find that
[ 𝟐]
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QUES
:
Using Mesh Analysis.
Mesh 1 contains a current source of 6 A. Hence, we can write current
equation for Mesh 1. Since direction of current source and mesh current
I1 are same,
Amp
(ii)
(iii
)
Ques: Using mesh analysis, determine the currents of the network in
Fig.
SUPERMESH
APPROACH
The single path now including the effects
of two mesh currents is referred to as the
path of a supermesh current.
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28
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NODAL ANALYSIS
A NODE is a point where two or more POINTS TO REMEMBER
circuit elements meet.
All the points or nodes
OR
A point at which two or more elements simply connected
have a common connection is called a together through a
resistance less wire or
NODE.
OR simply wire represents
A NODE is the point of connection the same node or point.
A JUNCTION can also be
between two or more branches.
If three or more branches join at a called as NODE but a NODE
node, then that node is called as may or may not be a
PRINCIPAL NODE or JUNCTION. JUNCTION.
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A BRANCH represents a single
element such as a voltage
source or a resistor.[A BRANCH
represents any two-terminal
element.]
OR
A BRANCH is defined as a single
Number of Nodes: 5
path in a network, composed of
Number of Principal Nodes: 2
one simple element and the
Number of Branches: 7
node at each end of that
element
[A BRANCH is that part of the
network which lies between two 31
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NODAL ANALYSIS
NODAL Analysis is used to find mainly the NODE VOLTAGE/S
(or Voltages of non – reference node) and other
variables if required.
KCL is adopted or implemented to find the NODE VOLTAGE/S
ALERT: All the points connected together by a
simple wire represents the same node or only one
single node.
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STEPS TO IMPLEMENT THE NODAL ANALYSIS
1.Determine the number of
4. Apply Kirchhoff’s current law at
nodes within the network. each node except the reference
2.Pick a reference node (or node. Assume that all unknown
datum node or ground node), currents leave the node for
and label each remaining each application of Kirchhoff’s
node with a subscripted current law. Each node is to be
value of voltage: V1 (or Va),V2 treated as a separate entity,
independent of the application
(or Vb) and so on.
of Kirchhoff’s current law to
3. [The negative of the battery is
the other nodes.
chosen as reference or datum node /
The node to which the maximum 5. Solve the resulting equations
number of elements are connected is for the nodal voltages.
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selected as datum or reference or
EXAMPLE: 1
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37
38
39
40
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20 8.48 V
V 6.06 V
2.304 1.696 0.605
A A A 0.606
A
0V
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EXAMPLE 2: Determine the node voltage and the current through all the
VA
branches Node Node
A A
1A
6Ω
12 Ω 6Ω
12 Ω
1A
+ 1A
24 V
- 24 V
+
-
Ground Node
0V
0V
Apply KCL at Node A
V V 24
1 A A or VA 2VA 48 12
12 6 VA 0
or 3VA 12 48 I12
V V 24 12
or A A 1
12 6 60 20
or VA 20 V I12
V V 24 3 12
or A A 1
12 6 or VA 20 V I12 1.666 Ampere
ANSWE
V 2VA 48
or A 1 R
12
Minus sign simply indicates that
direction of current should be opposite 43
to the assumed one.
EXAMPLE 3: Determine the node voltages and the current through all the branches in the
circuit
Node 1 Node 2
Ground Node
Steps 1 and 2: The network has three nodes, with the bottom node defined as the reference node (at
ground potential, or zero volts), and the other nodes as V1 and V2
Step 3: For node V1 the currents are defined as shown in Fig., and Kirchhoff’s current law at Node 1 is
applied:
Node
1
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Substituting values gives
or 0.375V1 0.25V2 6 (1)
For node V2 the currents are defined as shown in Fig., and Kirchhoff’s current law at Node 2 is applied:
Solving the two
Node 2
simultaneous equations i.e.
Eq. (1) and Eq. (2), V1 and
V2 are obtained as:
V1 = 37.82 Volt
V2 = 32.73 Volt
or 0.25V1 0.35V2 2 ( 2)
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Contd.
SOLUTION:
Circuit Diagram for
12 Ω Applying KCL at Node
1
Hint
2A
4A 2Ω
6Ω
Circuit Diagram for
Applying KCL at Node
2
ANSWERS
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EXAMPLE : Determine the node voltage and the current through all the
branches
V1 = 4 Volt
V2 = 8 Volt
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EXAMPLE : For the circuit shown in Figure, use nodal
analysis to find the node voltages at nodes 1 and 2.
V1 = 10.697 Volt
V2 = 7.5455 Volt
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EXAMPLE : For the circuit shown in Figure, use nodal analysis to
find the node voltages at nodes 1 and 2.
V1 = 4 Volt
V2 = 3 Volt
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EXAMPLE : Find V1 and V2 in the circuit shown in Figure
V1 = -2.9167 Volt
V2 = 7 Volt
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EXAMPLE : Find V1 and V2 in
the circuit shown in Figure
V1 = 11.4 Volt
V2 = 1.826 Volt
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EXAMPLE 7: Determine the node
EXAMPLE 4: Determine the node EXAMPLE 5: Determine the
voltage/s and the current
voltage/s and the current voltage across RL using nodal
through all the branches in the
through all the branches in the analysis. circuits shown below
circuits shown below
Va
Vc Vb
EXAMPLE 6: Determine the
voltage across vo using nodal
analysis.
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CONCEPT OF SUPERNODE
15 V
15 V 4Ω + -
5Ω 5Ω
20 Va + - Vb Va Vb
V
SUPERNODE
3Ω 10 10
Ω 20 Ω
20
V
V
0V
If an ideal voltage source is connected
between two non – reference nodes without
resistive element, the two non – reference
nodes collectively form SUPERNODE
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EXAMPLE: Find the node voltages and current in 5ohm resistor in
the circuit shown in Figure.
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2Ω
1Ω 5Ω
2Ω
OR
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EXAMPLE : Find V1 and V2 in the circuit shown in Figure
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Ques: For the circuit shown in
Fig., find the node voltages.
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Ques: For the circuit shown in
Fig., find the node voltages.
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Find the voltage across
0.33 Ω using both MESH
ANALYSIS and NODAL
ANALYSIS
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