This document provides an overview of network analysis concepts including: - Circuits are represented as graphs with branches and nodes. Circuit components reside in branches and connectivity in nodes. - Current, voltage, power, resistance, capacitors, inductors, and different types of sources such as voltage, current, dependent, and independent sources are defined. - Series and parallel connections are examined along with voltage and current division principles. - Kirchhoff's laws including nodal analysis and loop/mesh analysis techniques are introduced for solving circuit problems. - Active/passive elements as well as bilateral/unilateral elements are classified.

1. Network Analysis
UNIT-1
Dr Keshav Patidar

2. Network (Circuit)
• Represented as branches
and nodes in an undirected
graph.
• Circuit components reside
in the branches
• Connectivity resides in the
nodes
– Nodes represent wires
– Wires represent
equipotentials
• An electric circuit is an interconnection of elements.
• A circuit consists of a mesh of loops
Dr Keshav Patidar

3. Current
• The movement of charge.
• We always note the direction of the equivalent
positive charges, even if the moving charges are
negative.
• It is the time derivative of charge passing through a
circuit branch
• Unit is Ampere (A), is one Coulomb/second
• Customarily represented by i (AC) or I (DC).
dt
dq
i 
Dr Keshav Patidar

4. Voltage
• a difference in electric potential
– always taken between two points.
• It is a line integral of the force exerted by an
electric field on a unit charge.
• Customarily represented by u (AC) or U (DC)
or v and V alternativelly.
• The SI unit is the Volt [V].
Dr Keshav Patidar

5. Power
• Power is the product of voltage by current.
• It is the time derivative of energy delivered to or
extracted from a circuit branch.
• Customarily represented by P or S or W.
• The SI unit is the Watt [W].
Dr Keshav Patidar

6. AC vs. DC circuits
• Direct Current (DC) is a current
that remains constant with time is
called
• A common source of DC is a
battery.
• A current that varies sinusoidally
with time is called Alternating
Current (AC)
• Mains power is an example of AC
Dr Keshav Patidar

7. Resistance
When the same potential difference is applied across
different conductors, different currents flow.
Resistance R of a conductor is defined as the ratio of the
potential difference V applied across it to the current I
flowing through it.
unit: ohm
I
V
R 
Dr Keshav Patidar

8. •Most electric circuits use circuit
elements called resistors to control
the current in the various parts of
the circuit.
•Stand-alone resistors are widely
used.
– Resistors can be built into
integrated circuit chips.
• Values of resistors are normally
indicated by colored bands.
– The first two bands give the
first two digits in the
resistance value.
– The third band represents the
power of ten for the
multiplier band.
– The last band is the
tolerance.
Dr Keshav Patidar

9. Current, voltage and resistance
Current in a simple circuit will be larger if
• voltage of the supply is larger
• resistance in the circuit is smaller
where R is resistance, p is resistivity of the material, l is its length and A
is its cross-sectional area.
The same relationships apply in networks of identical
resistors.
A
l
R


Dr Keshav Patidar

10. Series Connection
• All components are connected end-to-end.
• Voltage drops add to total voltage.
• Due to all components goes the same (equal) current.
• Impedance (or simply resistance in DC) add to total impedance (resistance).
Dr Keshav Patidar

11. Parallel Connection
• All components are conected between the same two sets of electrically
common points.
• Currents add to total current.
• Voltage drop on the components are the same.
• Conductances (inverse of resistance) add to total conductance.
or
Dr Keshav Patidar

12. Resistor networks
Resistors in series
V = V1+ V2 [conservation of energy]
IR = IR1 + IR2
R = R1 + R2 R is always larger than any of R1, R2 etc
Resistors in parallel
I = I1 + I2 [conservation of charge]
V/R = V/R1 + V/R2
1/R = 1/R1 + 1/R2 R is always smaller than any of R1, R2 etc
Dr Keshav Patidar

13. Component characteristics
The electrical behaviour of a component is described by its
I-V graph.
For example:
I/V characteristic of a carbon resistor
I/V characteristic of a filament lamp
I/V characteristic of a semiconductor diode
Dr Keshav Patidar

14. Unlike the resistor which dissipates energy, ideal
capacitors and inductors store energy rather than
dissipating it.
Capacitor: In both digital and analog electronic
circuits a capacitor is a fundamental element. It
enables the filtering of signals and it provides a
fundamental memory element. The capacitor is an
element that stores energy in an electric field. The
circuit symbol and associated electrical variables for
the capacitor is shown on Figure 1.
Dr Keshav Patidar

15. The capacitor may be modeled as two conducting
plates separated by a dielectric as shown on Figure
2. When a voltage v is applied across the plates, a
charge +q accumulates on one plate and a charge –
q on the other.
Dr Keshav Patidar

16. The current flowing into the capacitor is the rate of
change of the charge across the capacitor plates
Dr Keshav Patidar

17. Dr Keshav Patidar

18. Capacitors connected in series and in parallel
combine to an equivalent capacitance. Let’s first
consider the parallel combination of capacitors as
shown on Figure
Dr Keshav Patidar

19. Dr Keshav Patidar

20. Next let’s look at the series combination of capacitors as shown on
Figure
Dr Keshav Patidar

21. Basic circuit elements - inductor
Inductance is the property whereby an inductor exhibits
opposition to the change of current flowing through it,
measured in henrys (H).
Dr Keshav Patidar

22. Basic circuit elements – inductor (2)
• The dependence between the current and the
voltage of the inductor is described by the
equations:
• The power stored by an inductor:
An inductor acts like a short circuit to dc (di/dt = 0) and its
current cannot change abruptly.
Dr Keshav Patidar

23. Example: There pure inductances are connected as
shown in Fig. What equivalent inductance Leq may
replace this circuit?
Dr Keshav Patidar

24. Ex: Find the equivalent
capacitance Ce of the
combination of
capacitors shown in Fig
Dr Keshav Patidar

25. Exercise: Find the total equivalent capacitance and total
energy stored if the applied voltage is 100 V for the
circuit as shown in Fig
Energy= 36900
Dr Keshav Patidar

26. Energy sources types
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27. Circuit Elements Ideal Independent
Voltage Source
• provides a specified voltage or current that is completely independent of
other circuit variables
• The voltage at the nodes is strictly defined by voltage of the source, the
current flow depends on the other elements in the circuit
The ideal voltage source is only a mathematical model.
Generally we can divide the voltage sources into three groups:
• Batteries
• Generators
• Supplies Dr Keshav Patidar

28. V-I Characteristics of Voltage Source
Dr Keshav Patidar

29. Circuit Elements Ideal independent
Current source
• The current flow in the branch is strictly defined by current of
the source, the voltage at the nodes of the source depends on
the other elements in the circuit
• The symbols used for AC current sources (similarly as for
voltage sources) are the same as for the DC current sources, but
described with noncapital letters (e.g. j(t)).
The ideal current source similarly
to ideal voltage source is only a
mathematical model. Dr Keshav Patidar

30. V-I Characteristics of Current Source
Dr Keshav Patidar

31. Circuit Elements – dependent
sources
• Ideal dependent
voltage source
• Ideal dependent
current source
The voltage defined by the
source depends on the voltage
or current determined in this
or other circuit
The current defined by the
source depends on the voltage
or current determined in this
or other circuit
Dr Keshav Patidar

32. Dependent or controlled sources are of the
following types:
(i) voltage controlled voltage source (VCVS)
(ii) current controlled voltage source (CCVS)
(iii) voltage controlled current source (VCCS)
(iv) current controlled current source (CCCS)
Circuit Elements – dependent
sources
Dr Keshav Patidar

33. The real voltage sources
The real voltage source can be
represented by ideal voltage
source in series with resistance Rs.
The real AC voltage source is
represented by ideal voltage source in
series with
impedance Zs. That impedance can have
resistive, inductive or capacitive character.
Dr Keshav Patidar

34. V-I Characteristics of Real(Practical) Voltage Source
Dr Keshav Patidar

35. The real current sources
The real current source can be
represented by ideal current source in
parallel with resistance Rs.
The real AC current source is
represented by ideal current source
with finite
impedance Zs placed across an ideal current
source. That impedance can have resistive,
inductive or capacitive character.
Dr Keshav Patidar

36. V-I Characteristics of Real Current Source
Dr Keshav Patidar

37. Source Conversion
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38. Determine the equivalent voltage source for the
current source shown in Fig.
Dr Keshav Patidar

39. Dr Keshav Patidar

40. Determine the equivalent current source for the voltage
source shown in Fig.
Dr Keshav Patidar

41. Dr Keshav Patidar

42. Find the voltage and current source equivalent
representation of the following network across open
terminals, as shown in Fig.
Dr Keshav Patidar

43. Dr Keshav Patidar

44. Example: Reduce the network shown in Fig., to a single loop
network by successive source transformation, to obtain the
current in the 12Ω resistor.
Dr Keshav Patidar

45. Dr Keshav Patidar

46. Dr Keshav Patidar

47. Voltage Division
The series circuit acts as a voltage divider. Since the same
current flows through each resistor, the voltage drops are
proportional to the values of resistors. Using this principle,
different voltages can be obtained from a single source, called a
voltage divider.
Dr Keshav Patidar

48. Dr Keshav Patidar

49. Example: voltage divider
Assume no current is drawn at the output
terminals in measuring Vout. Ohm’s Law
requires that VR1 = IR1 R1 and VR2 = IR2 R2,
which is also Vout. KCL says the current
leaving resistor R1 must equal the current
entering R2, or IR1 = IR2, so we can write
Vout = IR1 R2. KVL says the voltage around the loop including the battery
and both resistors is 0, therefore Vin = VR1 + Vout, or Vin = IR1 R1 + IR1 R2.
Thus, IR1 = Vin / (R1 + R2), and
Vout = Vin R2 / (R1 + R2).
R1
R2
Vin Vout
Dr Keshav Patidar

50. Example: What is the voltage
across the 10 Ω resistor in Fig.
Dr Keshav Patidar

51. Current Division:
In a parallel circuit, the current divides in all branches.
Thus, a parallel circuit acts as a current divider. The total
current entering into the parallel branches is divided into the
branches currents according to the resistance values. The
branch having higher resistance allows lesser current, and the
branch with lower resistance allows more current.
Dr Keshav Patidar

52. Dr Keshav Patidar

53. Exercise: Determine the current in the 10 Ω
resistance and find Vs in the circuit shown in Fig.
Dr Keshav Patidar

54. Determine the current in all resistors in the circuit
shown in Fig.
Dr Keshav Patidar

55. Kirchhoff ’s Circuit Laws
• Kirchhoff ’s circuit laws were first described in
1845 by Gustav Kirchhoff. They consist from
two equalities for the lumped element model
of electrical circuits. They describe the
current and voltage behaviour in the circuit.
Dr Keshav Patidar

56. Kirchhoff ’s First Law - Kirchhoff ’s
Current Law (KCL)
• The algebraic sum of currents in a network of conductors meeting at a
node is zero.
It can be described by the equation:
The currents flowing into the node (I1, I6) we describe as positive, the
currents flowing out the node (I2, I3, I4, I5) we describe as negative.
Dr Keshav Patidar

57. Ex: For the circuit shown in Fig., find the voltage
across the 10 resistor and the current passing
through it.
Dr Keshav Patidar

58. Kirchhoff ’s Second Law -
Kirchhoff ’s Voltage Law (KVL)
• The algebraic sum of the potential rises and
drops around a closed loop or path is zero.
where Ui describes both the potential
drops at the elements and the voltages
generated by sources.
To use the KVL one need to set up a rotation in the circuit. Potentials with
direction of the circuit have a positive sign, voltage opposite to the direction of
circulation of the circuit have a negative sign.
Dr Keshav Patidar

59. Example: For the circuit shown in Fig. , determine the
unknown voltage drop V1.
Dr Keshav Patidar

60. Node and Branch
A node is a point in a circuit where two or more
circuit elements join. A node is said to be an
essential node if it joins three or more
elements. Examples of nodes for Fig. are a, b,
c, d, e, f and g and examples of some
essential node of Fig. are b, c, e and g.
A branch is a path that connects two nodes. Those
paths that connect essential nodes without
passing through an essential node are known
as essential branches. Examples of branches
of Fig. are V1, R1, R2, R3, V2, R4, R5, R6,
R7 and I and some essential branches of Fig.
are c-a-b, c-d-e, c-f-g, b-e, e-g, b-g (through
R7 ), and b-g (through I ).
Dr Keshav Patidar

61. Broadly, network elements may be classified into
four groups, viz.
1. Active or passive
2. Unilateral or bilateral
3. Linear or nonlinear
4. Lumped or distributed
Dr Keshav Patidar

62. Active and Passive
Energy sources (voltage or current sources) are
active elements, capable of delivering power to
some external device. Passive elements are those
which are capable only of receiving power.
For example, ideal sources are active elements. A
passive element is defined as one that cannot supply
average power that is greater than zero over an
infinite time interval. Resistors, capacitors and
inductors fall into this category.
Dr Keshav Patidar

63. NETWORK ANALYSIS TECHNIQUES
Network analysis is the determination of the
response output of a network when the input
excitation is given.
There are two techniques of network analysis:
1. Nodal analysis
2. Loop or mesh analysis
Dr Keshav Patidar

64. Nodal Analysis
It is based on Kirchhoff’s current law (KCL). In this method, the unknown
variables are the node voltages. It is generally used when the circuit contains
several current sources.
Steps
(i) If there are ‘N’ number of nodes in a network, all nodes are labeled. One
node is treated as the datum or reference node (zero potential) and the other
node voltages are treated as unknowns to be determined with respect to this
reference.
(ii) KCL is written at each node in terms of node voltages.
(a) KCL is applied at N -1 of the N nodes of the circuit using assumed
current directions, as necessary.
This will create N -1 linearly independent equations, known as node
equations.
Dr Keshav Patidar

65. For the network shown in Fig. , apply Kirchhoff ’s current law
and write the node equations.
Dr Keshav Patidar

66. Loop or Mesh Analysis
It is based on Kirchhoff’s voltage law (KVL). In this method, the unknown
variables are the loop currents. It is generally used when the circuit
contains several voltage sources.
Steps
(i) If there are ‘N’ number of loops/meshes in a network, all loops are
labeled.
(ii) KVL is written at each loop/mesh in terms of loop/mesh currents.
Loop currents are those currents flowing in a loop; they are used to define
branch currents.
(a) For N independent loops, a total of N equations are written using KVL
around each loop. These equations are known as loop/mesh equations.
Dr Keshav Patidar

67. Write the mesh equations for the circuit shown in Fig
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68. Bilateral and Unilateral
In the bilateral element, the voltage-current relation
is the same for current flowing in either direction.
In contrast, a unilateral element has different
relations between voltage and current for the two
possible directions of current. Examples of
bilateral elements are elements made of high
conductivity materials in general. Vacuum diodes,
silicon diodes, and metal rectifiers are examples of
unilateral elements.
Dr Keshav Patidar

69. Dr Keshav Patidar

70. Linear and Nonlinear Elements
An element is said to be linear, if its voltage-
current characteristic is at all times a straight line
through the origin. For example, the current
passing through a resistor is proportional to the
voltage applied through it.
Dr Keshav Patidar

71. Lumped and Distributed
Lumped elements are those elements which are very
small in size and in which simultaneous actions takes
place for any given cause at the same instant of time.
Typical lumped elements are capacitors, resistors,
inductors and transformers. Generally the elements are
considered as lumped when their size is very small
compared to the wave length of the applied signal.
Dr Keshav Patidar

72. Distributed elements, on the other hand, are those
which are not electrically separable for analytical
purposes. For example, a transmission line which
has distributed resistance, inductance and
capacitance along its length may extend for
hundreds of miles.
Dr Keshav Patidar

73. AC resistor circuits
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74. AC capacitor circuits
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75. AC capacitor circuits
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76. AC inductor circuits
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77. AC inductor circuits
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78. Dr Keshav Patidar