Network Analysis & Synthesis BEC 303

Unit-I
Nature of Electricity

 Electricity is the most common form of energy. Electricity is used for various applications
such as lighting, transportation, cooking, communication, production of various goods in
factories and much more.

 For observing nature of electricity, it is necessary to study the structure of matters.

 Matters Molecules Atoms Electrons

Molecules consists of similar atoms Elements

Molecules consists of dissimilar atoms Compounds

 Structure of atom: An atom consists of one central nucleus. The nucleus is made up of
positive protons and charge less neutrons. This nucleus is surrounded by numbers of orbital
electrons. Each electron has a negative charge of – 1.602 × 10 – 19 Coulomb and each proton in
the nucleus has a positive charge of +1.602 × 10 – 19 Coulomb.

 Because of the opposite charge there is some attraction force between the nucleus and orbiting
electrons. Electrons have relatively negligible mass compared to the mass of the nucleus. The
mass of each proton and neutrons is 1840 times the mass of an electron.

 As the modulus value of each electron and each proton are same, the number of electrons is
equal to the number protons in an electrically neutral atom. An atom becomes positively
charged ion when it loses electrons and similarly an atom becomes negative ion when it gains
electrons.

 Atoms may have loosely bonded electrons in their outermost orbits. These electrons require a
very small amount of energy to detach themselves from their parent atoms. These electrons
are referred as free electrons which move randomly inside the substance and transferred from
one atom to another.

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Figure 1.1 Structure of Atom

 The movement of these free electrons can easily be directed to a particular direction if the
electrical potential difference is applied across the piece of these materials. Because of plenty
of free electrons these materials have good electrical conductivity. These materials are
referred as good conductor.

 The drift of electrons in a conductor in one direction is known as the current. Actually
electrons flow from lower potential (-Ve) to higher potential (+Ve) but the general
conventional direction of current has been considered as the highest potential point to lower
potential point, so the conventional direction of current has been just opposite of the direction
of flow of electrons.

 In non-metallic materials, such as glass, mica, slate, porcelain, the outermost orbit is
completed and there is almost no chance of losing electrons from its outermost shell. Hence
there is hardly any free electron present in this type of material.
Hence, these materials cannot conduct electricity in other words electrical conductivity of
these materials is very poor. Such material are known as non-conductor or electrical insulator.

1.1. Some Important Definitions

(a) Current

Figure 1.2 Concept of Electric Current

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 Flow of electron in closed circuit is called current.

 Amount of charge passing through the conductor in unit time also called current.

 Unit of current is charge/second or Ampere (A).

𝑄
𝐼=
𝑡

Where, I =Current

Q = Charge

t = Time

(b) Potential or Voltage

 The capacity of a charged body to do work is called potential.

 Unit of potential is joule/coulomb or Volt (V).

𝑊
𝑉=
𝑄

Where, V = Potential or Voltage

W = Workdone

Q = Charge

(c) Potential difference

Figure 1.3 Potential Differences

 The difference of electrical potential between two charged bodies is called potential
difference.

 Unit of Potential Difference is Volt (V).

 If potential of body A is +12V and potential of body B is +7V then potential difference is +5V.

i.e. (+12V) - (+7V) = +5V

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(d) Electro Motive Force (emf)

 The force is required to move electron from negative terminal to positive terminal of
electrical source in electrical circuit is called emf.

 Unit of emf is volt (V).

 Emf is denoted as ε.

(e) Energy

 Ability to do work is called energy.

 Unit of energy is Joule or Watt-sec or Kilowatt-hour (KWh).

 1KWh is equal to 1 Unit.

𝑉 𝑡
𝑊 = 𝑃𝑥𝑡 = 𝑉𝐼𝑡 = 𝐼 𝑅𝑡 =
𝑅
Where, W =Energy

P =Power

t =Time

(f) Power

 Energy per unit in time is called power.

 Unit of Power is Joule/Second or Watt (W).

𝑊
𝑃=
𝑡

(g) Resistance

 Property of a material that opposes the flow of electron is called resistance.

 Unit of resistance is Ohm (Ω).

𝑉
𝑅=
𝐼

Where, R = Resistance

(h) Conductance

 Property of a material that allows flow of electron.

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 It is reciprocal of resistance.

 Unit of conductance is (Ω-1) or mho or Siemens(S).

1
𝐺=
𝑅

Where, G = Conductance

(i) Resistivity or Specific Resistance

 Amount of resistance offered by 1m length of wire of 1m 2 cross-sectional area.

 Resistivity is denoted as a ρ.

 Unit of Resistivity is Ohm-meter (Ω-m).

𝑙
𝑅=𝜌
𝐴

Where, R = Resistance

ρ = Resistivity

𝑙 = 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑤𝑖𝑟𝑒

A = Area of cross-sectional of wire

(j) Conductivity

 Ability of a material to allow flow of electron of a given material for 1 m length & 1 m 2 cross-
1
sectional area is called conductivity. Unit of conductivity is Ω-1m-1 or Siemens m- .

1
𝜎=
𝜌

Where, σ = Conductivity

1.2. Concept of Network

 Any combination & interconnection of network elements like R, L & C or electrical energy
sources are known as a network.

 However, a closed energized network is known as a circuit.

 It may be noted that network need not contain an energy source, but a circuit must contain at
least one energy source.
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 Note: All the circuits are networks, but vice-versa are not necessarily true.

1.3. Basic Circuit Elements

1.4. Types of Electrical Energy Sources (Active Elements)

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1.5. Ideal & Practical Sources

a) Voltage Source

 A voltage source is a two-terminal device whose voltage at any instant of time is constant
and is independent of the current drawn from it. Such a voltage source is called an Ideal
Voltage Source and have zero internal resistance.

 Practically an ideal voltage source cannot be obtained.

 Sources having some amount of internal resistances are known as Practical Voltage Source.
Due to this internal resistance; voltage drop takes place, and it causes the terminal voltage to
reduce. The smaller is the internal resistance (r) of a voltage source, the more closely it is to
an Ideal Source.

 The symbolic representation of the ideal and practical voltage source is shown below.

Figure 1.4 Symbolic representation of voltage source

Figure 1.5 shown below shows the circuit diagram and characteristics of an ideal voltage
source:

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Figure 1.5 Ideal voltage source

Figure 1.6 shown below gives the circuit diagram and characteristics of Practical Voltage Source:

Figure 1.6 Practical voltage source

b) Current Source
 The current sources are further categorized as Ideal and Practical current source.

 An ideal current source is a two-terminal circuit element which supplies the same current to
any load resistance connected across its terminals. It is important to keep in mind that the
current supplied by the current source is independent of the voltage of source terminals. It has
infinite resistance.

 A practical current source is represented as an ideal current source connected with the
resistance in parallel. The symbolic representation is shown below:

Figure 1.7 Symbolic representation of current source

Figure 1.8 shown below, show its characteristics.

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Figure 1.8 Ideal current source

Figure 1.9 shown below shows the characteristics of Practical Current Source.

Figure 1.9 Practical current source

1.6. Linear Elements & Non-linear Elements (Passive Elements)

 A linear circuit is one whose parameters are constant with the time also they do not change
with voltage or current & circuit follows Ohm’s Law. V-I characteristics of a linear circuit is a
straight line.

Resistors, capacitors, Transformers, and Inductors are linear components.

 A Non-Linear circuit or non-linear element is one whose parameters vary with the time and
change with voltage and current and also they don’t follow the Ohm’s Law. V-I
characteristics of the non-Linear circuit is Not a straight line.

Transistor, Diodes are Non -Linear elements.

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Figure 1.10 Linear & Non-linear Elements Curve

(a) Resistor

 It is a two terminal electrical or electronic component that resists an electric current by

producing a voltage drop between its terminals in accordance with Ohm’s law.

Figure 1.11 Resistor & Conductor

 Resistance is the property of a material which opposes the flow of an electric current. It is
measured in Ohms (Ω).

 Value of resistance of conductor is

 Proportional to its length.

 Inversely proportional to the area of cross section.

 Depends on nature of material.

 Depends on temperature of conductor.

𝑙
𝑅=𝜌
𝐴

(b) Inductor

 An inductor is an element which stores energy in the form of magnetic field.

 The property of the coil of inducing emf due to the changing flux linked with it is known as an
inductance of the coil.

 Inductance is denoted by L and it is measured in Henry (H).

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Figure 1.12 Inductor

 Value of inductance of coil is

 Directly proportional to the square of number of turns.

 Directly proportional to the area of cross section.

 Inversely proportional to the length.

 Depends on absolute permeability of magnetic material.

Where, L =Inductance of coil

N= Number of turns of coil
Φ = Flux link in coil
F = Magneto motive force (MMF)
I = Current in the coil
l = Mean length of coil
μ0 = Permeability of free space
μr = Relative permeability of magnetic material
A = Cross sectional area of magnetic material
(c) Capacitor

 Capacitor is an element which stored energy in the form of charge.

 Capacitance is the capacity of capacitor to store electric charge.

 It is denoted by C and measured in Farad (F).

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Figure 1.13 Capacitor

 Value of capacitance is

 Directly proportional to the area of plate.

 Inversely proportional to distance between two plates.

 Depends on absolute permittivity of medium between the plates.

ℰ𝐴
𝐶=
𝐷
ℰ =ℰoℰr
Where, C=Capacitance of capacitor
A =Cross sectional area of plates
d =Distance between two plates
ε = Absolute Permittivity
ε0 = Permittivity of free space
εr = Relative permittivity of dielectric material

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1.7. Unilateral and Bilateral Elements

 Unilateral Elements: When element properties & characteristics dependent on the direction
of current then the element is called as unidirectional or unilateral element. For e.g. diode,
transistor, etc.

Figure 1.14 Unilateral element

Note: If |I| = |I’|, then element is called as unilateral or unidirectional element.
 Bilateral Elements: When element properties & characteristics independent on the direction
of current then the element is called as bidirectional or bilateral element. For e.g. R, L & C are
bilateral elements.

Figure 1.15 Bilateral element

Note: If |I| = |I’|, then element is called as bilateral or bidirectional element.
Note:
1) When V/I ratio is positive in both co-ordinates then the element is passive otherwise active.
2) Every linear element must obey the bilateral property. However, vice-versa is not necessarily
true.
3) It may be noted that bilateral curve identical in opposite plane not in adjacent plane.
4) An element is said to be linear whose V-I characteristics follows only one equation of straight
line passing through origin for all times.

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Figure 1.16 V-I Curve

1.8. Ohm’s law and its limitations

 Ohm’s law states that current flowing through the conductor is directly proportional to the
potential difference applied to the conductor, provided that no change in physical conditions, i.e.
temperature, etc.

V = IR

𝑉
𝑅=
𝐼

Where R is constant which is called resistance of the conductor.

Figure 1.17 Change in current w.r.t change in voltage for conducting material

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Limitations of Ohm’s Law:

 It cannot be applied to non-linear device e.g. Diode, Zener diode etc.

 It cannot be applied to non-metallic conductor e.g. Graphite, Conducting polymers

 It can only be applied in the constant temperature condition.

1.9. Source Transformations
 A voltage source with a series resistor can be converted into an equivalent current source with
a parallel resistor. Conversely, a current source with a parallel resistor can be converted into a
voltage source with a series resistor.

 Open circuit voltages in both the circuits are equal and short circuit currents in both the circuit
are equal. Source transformation can be applied to dependent source as well.

Figure 1.18 Source transformation

Network Simplification Techniques

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Figure 1.19 Sources Combinations

1.10. Kirchhoff’s law

(a) Kirchhoff’s current law (KCL)
 Statement: “Algebraic sum of all current meeting at a junction is zero”
Let, Suppose

 Branches are meeting at a junction ‘J’

 Incoming current are denoted with (+ve) sign

 Outgoing currents are denoted with (-ve) sign

Figure 1.20 Kirchhoff’s law circuit diagram

(b) Kirchhoff’s voltage law (KVL)

 Statement: “Algebraic sum of all voltage drops and all emf sources in any closed path is zero”
 Let, Suppose
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 Loop current in clockwise or anticlockwise direction

 Circuit current and loop current are in same direction than voltage drop is denoted by (-ve)
sign.

 Circuit current and loop current are in opposite direction than voltage drop is denoted by
(+ve) sign.

 Loop current move through (+ve) to (-ve) terminal of source than direction of emf is (-ve).

 If Loop current move through (-ve) to (+ve) terminal of source than direction of emf is
(+ve).

Figure 1.21 Sign convention for Kirchhoff’s voltage law

1.11. Series & parallel combination of resistor

Series combination of resistor Parallel combination of resistor

Figure 1.22 Series combination of resistor

Figure 1.23 Parallel combination of resistor

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Value of equivalent resistance of series circuit is

bigger than the biggest value of individual Value of equivalent resistance of parallel
resistance of circuit. circuit is smaller than the smallest value of
individual resistance of circuit

1.12. Voltage divider rule & current divider rule

Voltage divider rule Current divider rule

Figure 1.24 Voltage divider circuit Figure 1.25 Current divider circuit

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1.13. Network Analysis

 Choice between loop & nodal methods of analyzing a network

1) The mesh method is generally used for circuits having many series connected elements.
2) Nodal analysis is more suitable for circuits having many parallel connected elements.
3) Usually if node voltages are required, nodal analysis is used and if branch currents are
required, loop analysis is used.
 Definitions of certain terms
1) Node: Any point in a circuit where the terminals of two or more elements are connected
together.
2) Branch: A branch is a part of the circuit which extends from one node to another. A
branch may contain one element or several elements in series.
3) Essential node (junction): If three or more elements are connected together at a node,
then that node is called essential node or junction.
4) Essential branch: A path which connects two essential nodes without passing through
an essential node is called essential branch.
5) Mesh & Loop: A loop or mesh denotes a closed path obtained by starting at node and
returning back to the same node. A mesh is a smallest closed path which does not
contain any other closed path within it.
Note- All the mesh are loop, but vice-versa not necessarily true.
1.14. Mesh Analysis

Figure 1.27 Mesh analysis network

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 Mesh: It is defined as a loop which does not contain any other loops within it.

 The current in different meshes are assigned continues path that they do not split at a junction
into a branch currents.

 Basically, this analysis consists of writing mesh equation by Kirchhoff’s voltage law in terms
of unknown mesh current.

 Steps to be followed in mesh analysis:

 Identify the mesh, assign a direction to it and assign an unknown current in it.

 Assigned polarity for voltage across the branches.

 Apply the KVL around the mesh and use ohm’s law to express the branch voltage in term of
unknown mesh current and resistance.

 Solve the equations for unknown mesh current.

Loop 1

Figure 1.28 Mesh analysis network for loop 1

………………………… (i)

Loop 2

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Figure 1.29 Mesh analysis network for loop 2

…………………. (ii)

Loop 3

Figure 1.30 Mesh analysis network for loop 3

………………….. (iii)
Now, from equations (i), (ii) & (iii) we can calculate all branch current with the help of
calculator.
1.15. Nodal analysis

Figure 1.31 Node analysis network

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 Node: Node refers to any point on circuit where two or more circuit elements meet.

 Node analysis based on Kirchhoff’s current law states that algebraic summation of currents
meeting at junction is zero.

 Node C is taken as reference node in this network. If there are n nodes in any network, the
number of equation to be solved will be (n-1).

 Node A, B and C are shown in given network and their voltages are VA, VB and VC. Value of
node VC is zero because VC is reference node.

 Steps to follow in node analysis:

 Consider node in the network, assign current and voltage for each branch and node
respectively.

 Apply the KCL for each node and apply ohm’s law to branch current.

 Solve the equation for find the unknown node voltage.

 Using these voltages, find the required branch currents.

Node A

Figure 1.32 Node analysis network for node A

………………. (i)
Node B

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Figure 1.33 Node analysis network for node B

……………. (ii)
 One can easily find branch current of this network by solving equation (i) and (ii), if V1 ,
V2 and all resistance value are given.

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