UNIT I

Introduction to Electrical Circuits

Syllabus: Basic Concepts of passive elements of R, L, C and their V-I relations, Types of electrical elements,
Sources (dependent and independent), Kirchoff’s laws, Network reduction techniques (series, parallel, series
- parallel, star-to-delta and delta-to-star transformation), source transformation technique, nodal analysis and
mesh analysis to DC networks with dependent and independent voltage and current sources., node and
mesh analysis, Super node and Super mesh analysis.

Introduction to Circuit concept: An electric circuit is formed by interconnecting components having different
electric properties.
Difference between circuit and network: “An electrical network is an interconnection of electrical elements
such as resistors, inductors, capacitors, transmission lines, voltage sources, current sources, and switches”.
An electrical circuit is a network that has a closed loop, giving a return path for the current.
Note: All circuits are networks but a network is a connection of two or more components, and may
not necessarily be a circuit. (For example blue coloured is a network & circuit but below red colour is a network
but not a circuit)

1.1 Basic Definitions in Electrical Circuits & Networks

a) Voltage: Voltage (or potential difference) is the energy required to move charge from one point to the
other, measured in volts (V).
𝑑𝑊
Mathematically, Voltage (V) = , volts or Joules per coulomb
𝑑𝑞

It is the potential difference between two points which are connected through element(s) (or) the total work
per moving a unit charge associated with the motion of charge between two points is called voltage.

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b) Current: Current can be defined as the motion of charge through a conducting material, measured in
Ampere (A). (Or) The current can also be defined as the rate of charge passing through a point in an electric
𝑑𝑞
circuit. Mathematically, Current (I) =
𝑑𝑡
c) Power: It is the time rate of expending or absorbing energy, measured in watts (W). Power, is denoted by
the letter P.

d) Energy: Energy is the capacity to do work, measured in joules (J).

Energy (E) = Power * time, Watt-Seconds or Joules
Although the unit of energy is the joule, when dealing large amounts of energy, the unit used is the kilowatt
hour (kWh) where 1 Wh=3600 J. (1unit of electrical energy = 1 kWh = 3600000 J)
e) Ohm’s law: Ohm's law states that the current through a conductor between two points is directly
proportional to the voltage across the two points. Therefore, V = RI where R is a constant called resistance.
R depends on the dimensions of the conductor and also on the material of the conductor.
Limitation of Ohm’s law:
 Ohm's law is applicable when the temperature of the conductor is constant. Resistivity changes with
temperature.
 The relation between voltage and current depends on the sign of voltage.
 It does not apply to semiconductors, which do not have a direct current-voltage relationship.
1.2 Types of electrical elements
a) Active and passive elements: An electrical element which supplies electrical energy is called an active
element. Examples: Voltage sources and current sources (Say a Battery)
The passive elements receive energy from the sources. The passive elements are the elements with
resistance, the inductance and the capacitance.
b) Linear and Non-Linear Elements: Linear elements show the linear characteristics of voltage & current.
That is its voltage-current characteristics are at all-times a straight-line through the origin.
Resistors, inductors and capacitors are the examples of the linear elements and their properties do not
change with a change in the applied voltage and the circuit current.
Nonlinear element’s V-I characteristics do not follow the linear pattern i.e. the current passing through it does
not change linearly with the linear change in the voltage across it. Examples are the semiconductor devices
such as diode, transistor.
c) Bilateral and Unilateral: If an element provides same amount of resistance for the current flow in both
directions, then it is called as bilateral otherwise it is unilateral element.
Example for bilateral elements are resistor, inductor and capacitor in various electrical and electronic circuits
and diode is an example for unilateral element.
d) Lumped and distributed elements: If resistance, inductance and capacitance of an element are
distributed equally throughout its length, then that element is considered as distributed element. Example for
distributed element is a transmission line has distributed parameters along its length and may extend for

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hundreds of miles. If resistance, inductance and capacitance of an element are lumped or concentrated for
a particular length of element, then that element is considered as lumped element.
Example for distributed element is transmission lines
Example for lumped element is resistor, inductor, capacitor in various electrical and electronic circuits
1.3 R L C Parameters
1.3.1 Resistor: Resistance is that property of a circuit element which opposes the flow of electric current and
in doing so converts electrical energy into heat energy.
When current flows through any resistive material, heat is generated by the collision of electrons with other
atomic particles. The power absorbed by the resistor is converted to heat and is given by the expression
P= VI = I2R Watts
where I is current through resistor in amps, and v is the voltage across the resistor in volts.
𝑡 𝑣2
Energy lost in a resistance in time t is given by W =∫
0
𝑝 𝑑𝑡 = 𝑝𝑡 = 𝑖 2 𝑅𝑡 = t
𝑅
Example for resistors are like water immersion electric heater, iron box which possess more resistance
nature.
Properties of resistor: 1. It is linear, bilateral and passive element
2. Resistor never stores energy, only dissipates(wastes) energy in the form of heat.
3. It allows sudden changes in voltage and current.

1.3.2 Inductor: A wire of certain length, when twisted into a coil becomes a basic conductor.
Symbolic representation of inductor is

ɸ
L=
𝐼
The amount of flux produced in a coil for unit (or 1 Amp) current flow through the coil. Units are Henrys.

di
Induced Voltage in inductor is given by VL = L
dt
VL = Voltage across inductor in volts & i = Current through inductor in amps
1
From above equation, we can have di = v dt,
L
t 1 t
Integrating both sides, ∫
0
di = ∫0 vdt
L

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 𝑑𝑖
Power absorbed by the inductor P = vi = Li
𝑑𝑡
𝑡 𝑑𝑖 𝐿𝑖 2
Energy stored by the inductor W=∫
0
𝑃 𝑑𝑡 = ∫0𝑡 𝐿𝑖 𝑑𝑡
dt = ,
2
𝐿𝑖 2
Therefore W = Joules
2

Properties of inductor
1. It is a linear, bilateral and passive element.
di
2. We have VL = L dt . The induced voltage across an inductor is zero if the current through it is constant

(i.e., DC voltage). That means an inductor acts as short circuit to dc.

3. A small change in current within zero time (dt = 0) gives an infinite voltage across the inductor which
is physically not at all feasible. Hence inductor does not allow sudden changes in current.
4. A pure inductor never dissipates energy; it only stores it in magnetic form.

Worked example: The current in a 2H inductor raises at a rate of 2A/s. Find the voltage across the inductor
& the energy stored in the magnetic field at after 2sec.
𝑑𝑖 1 1
V = L 𝑑𝑡 =2X2 = 4V and W= 2
Li2 = 2
X 2 X (4)2 = 16 Joules

1.3.3 Capacitor: A capacitor is a passive element designed to store energy in its electric field.

When a voltage source V or Vc is connected to the capacitor, the source deposits a positive charge q on one
plate and a negative charge −q on the other. The capacitor is said to store the electric charge. The amount
of charge stored, represented by q, is directly proportional to the applied voltage v so that:
Hence, Charge q is proportional to V, q=C.V
𝑞
therefore, C =
𝑉
“The amount of charge stored in an element for one voltage applied across that element is called the
capacitance” of that element.
𝑑𝑞 𝑑𝑣 𝒅𝒗
The current flowing in the circuit is rate of flow of charge i = =C ∴𝒊= 𝐂
𝑑𝑡 𝑑𝑡 𝒅𝒕

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 𝑑𝑞 𝑑𝑣 𝑑𝑣 𝑖 𝟏
i=
𝑑𝑡
=C
𝑑𝑡
,
𝑑𝑡
=𝑐 , V=
𝒄
∫ 𝒊𝒅𝒕
𝑑𝑣
The power absorbed by the capacitor P = vi = vc
𝑑𝑡
𝑡 𝑡 𝑑𝑣
Energy stored in the capacitor W= ∫0 𝑃𝑑𝑡 = ∫0 𝑉𝐶 𝑑𝑡
dt
𝟏
𝑡
= C∫0 𝑣𝑑𝑣 = 𝒄𝒗𝟐 Joules
𝟐
This energy is stored in the electric field set up by the voltage across capacitor.
Properties of capacitor
1. It is a linear, bilateral and passive element.
dv
2. We have i = C dt . The current flow through capacitor is zero if the voltage applied to capacitor constant

(i.e., DC voltage). That means an inductor acts as short circuit to dc.

3. A small change in voltage within zero time (dt = 0) gives an infinite current through the capacitor which
is physically not at all feasible. Hence capacitor does not allow sudden changes in voltage.
4. A pure capacitor never dissipates energy; it only stores it in electric form.

Note: Real appearance of passive element (R, L and C) components

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V-I Relation of circuit elements
Circuit elements Voltage(V) Current(A) Power(W)

𝑉
Resistor R (Ohms Ω) V=RI I=𝑅 P =i2R

1
I =𝐿 ∫ 𝑣𝑑𝑡 +io

𝑑𝑖 where i0 is the initial 𝑑𝑖

Inductor L (Henry H) V=L𝑑𝑡 P =Li 𝑑𝑡
current through
inductor
1
I= ∫ 𝑖𝑑𝑡 +v0
𝐶
𝑑𝑣 𝑑𝑣
Capacitor C (Farad F) where v0 is the initial I =C 𝑑𝑡 P =C𝑉 𝑑𝑡

voltage across capacitor

Example problems:
1. A current waveform is applied to a 2 H inductor. Draw the voltage waveform for shown below.

2. A voltage waveform shown in below figure, is applied to the capacitor. Draw the current waveform.

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1.4 Types of sources

Figure: Types of electrical sources

An independent voltage source maintains a voltage (fixed or varying with time) which is not affected by any
other quantity. Similarly, an independent current source maintains a current (fixed or time-varying) which is
unaffected by any other quantity. But in a dependent source is a voltage source or a current source whose
value depends on a voltage or current somewhere else in the network.
1.4.1 Independent sources: Different independent sources are discussed below
Ideal voltage source: If the voltage (terminal or output voltage) of a voltage source is constant irrespective
of current flowing through that source, then it is called as ideal voltage source, i.e. voltage drop in the source
is zero, as source resistance is zero (voltage drop = current * source resistance= zero).

Figure: Ideal voltage source & practical voltage source

Practical voltage source: If the voltage (terminal or output voltage) of a voltage source not constant and
decreases with increased current flow through that source, then it is called as practical voltage source, i.e.
voltage drop in the source is not zero and increases with current, as source resistance is not zero
Note: Practical voltage source is represented by its voltage in series with its source resistance (as shown
above). For ideal voltage source, source resistance is zero.
Ideal current source: If the current supplied (to output or terminals) by of a current source is constant
irrespective of voltage across that source, then it is called as ideal current source.
As ideal current source is having infinite source resistance in parallel with it, no current flows through that
source resistance (i.e., no current diversion at source to ground or negative), and hence it supplies constant
current.
Practical current source: If the current supplied (to output or terminals) by of a current source is not constant
& decreases with increasing of voltage across that source, then it is called as practical current source.
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As practical current source is having finite source resistance in parallel with it, some (leakage) current flows
through that source resistance (i.e. some of source current is diverted through that source resistance to
ground or negative), and hence it supplies decreased current.

1.4.2 Dependent sources:

The dependent, or controlled, source, in which the source quantity is determined by a voltage or current
existing at some other location in the system being analysed. Sources such as these appear in the equivalent
electrical models for many electronic devices, such as transistors, operational amplifiers, and integrated
circuits.

Figure: The four different types of dependent sources: (a) current-controlled current source; (b) voltage-
controlled current source; (c) voltage-controlled voltage source; (d) current controlled voltage source, where
K is a constant

Terminology in a network:
1. A point at which two or more elements have a common connection is called a node.
2. If no node was encountered more than once, then the set of nodes and elements that connected
through is defined as a path.
3. If the node at which we started is the same as the node on which we ended, then the path is, by
definition, a closed path or a loop.
4. Branch is a single path in a network, composed of one simple element or elements and the node at
each end of that element.
5. A 'mesh' is any closed path in a given circuit that does not have any element (or branch) inside it

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Example: Referring to the circuit depicted below, count the number of (a) nodes; (b) elements; (c) branches.

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1.5 KIRCHHOFF’S LAWS: (Most useful laws in analyzing various electrical and electronic circuits)
 Ohm’s law by itself is not sufficient to analyse circuits. However, when it is coupled with Kirchhoff’s
two laws, we have a sufficient, powerful set of tools for analysing a large variety of electric circuits.
 Kirchhoff’s first law is based on the law of conservation of charge, which requires that the algebraic
sum of charges within a system cannot change.
 Kirchhoff’s second law is based on the law of conservation of energy, which requires that the algebraic
sum of energies within a system cannot change

1.5.1 KIRCHHOFF’S CURRENT LAW: Kirchhoff’s current law (abbreviated KCL), simply states that “the
algebraic sum of the currents entering (or leaving) any node is zero.”
Consider the following example:

According to KCL, the algebraic sum of the four currents entering the node must be zero:

However, the law could be equally well applied to the algebraic sum of the currents leaving the node:

We can equate the sum of the currents having reference arrows directed into the node to the sum of those
directed out of the node:

The above equation simply states that the “sum of the currents going in must equal the sum of the currents
going out”.

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Example: For the circuit in Fig., compute the current through resistor R3 (marked as i) if it is known that the
voltage source supplies a current of 3 A. [Answer: 6Amps]

1.5.2 KIRCHHOFF’S VOLTAGE LAW: Current is related to the charge flowing through a circuit element,
whereas voltage is a measure of potential energy difference across the element.
Kirchhoff’s voltage law (abbreviated KVL) states that the algebraic sum of the voltages around any closed
path (loop or mesh) is zero.

According to KVL

We can apply KVL to a circuit around the closed path in a clockwise direction and writing down directly the
voltage of each element whose (+) terminal is entered, and writing down the negative of every voltage first
met at the (−) sign. Applying this to the single loop

By the rearrangement of above equation, we will get

The above equation simply states that “in a closed path sum of the voltage rises (source) must equal the sum
of the voltage drops (passive element)”.
Example: By using KVL find the value of Vx and also determine current ix [Answer: 12Volts & 0.12Amps]

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1.6 Network reduction techniques
1.6.1 Series and parallel connection of passive elements:
a) Total or equivalent resistance of circuit if ‘N’ resistors connected in series:
Consider the following circuit with N resistor connected in series, the equivalent resistance can be calculated
as shown below:

By applying KVL, we get

Now by applying ohm’s law, we get

Now compare this result with the simple equation applying to the equivalent circuit shown above

Thus, the value of the equivalent resistance for N series resistors is

Therefore, we are able to replace a two-terminal network consisting of N series resistors with a single two-
terminal element Req that has the same v-i relationship.

b) Total or equivalent resistance of circuit if ‘N’ resistors connected in parallel:

Figure: (a) A circuit with N resistors in parallel. (b) Equivalent circuit.

Similar simplifications can be applied to parallel circuits. A circuit containing N resistors in parallel, as in above
circuit leads to the KCL equation:

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Now by applying Ohm’s law, we get

Thus,

which can be written as

or, in terms of conductance’s, as

The simplified (equivalent) circuit is shown in figure. A parallel combination is routinely indicated by the
following shorthand notation:

The special case of only two parallel resistors is encountered fairly often, and is given by

or more simply

c) Total or equivalent inductance if ‘N’ inductors are connected in series:

Consider an ideal voltage source applied to the series combination of N inductors, as shown below and
a single equivalent inductor, with inductance Leq, which may replace the series combination so that the
source current i (t) is unchanged.

Figure: (a) A circuit containing N inductors in series. (b) The desired equivalent
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Applying KVL to the original circuit,

But for the equivalent circuit we have

and thus the equivalent inductance is

The inductor which is equivalent to several inductors connected in series is one whose inductance is the
sum of the inductances in the original circuit. This is exactly the same result we obtained for resistors in
series.

d) Total or equivalent inductance if ‘N’ inductors are connected in parallel:

Consider an ideal current source applied to the parallel combination of N inductors, as shown below and a
single equivalent inductor, with inductance Leq, which may replace the parallel combination so that the source
voltage v(t) is unchanged.

Figure: (a) The parallel combination of N inductors. (b) The equivalent circuit,
where Leq = [1/L1 + 1/L2 + · · · + 1/LN]−1.
By applying KCL,we get is(t) = i1(t) + i2(t) + …. iN(t)
1 1 1 1 1 1
=
𝐿1
∫ 𝑉𝑑𝑡 + 𝐿2
∫ 𝑉𝑑𝑡 + ⋯ + 𝐿 ∫ 𝑉𝑑𝑡 = (𝐿 + 𝐿 + ⋯ 𝐿𝑁
) ∫ 𝑉𝑑𝑡
𝑁 1 2
1
For equivalent circuit is(t) =
𝐿𝑒𝑞
∫ 𝑉𝑑𝑡

1 1 1 1
∴ = ( + +⋯ )
𝐿𝑒𝑞 𝐿 𝐿 1 2 𝐿𝑁

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e) Total or equivalent inductance if N capacitors are connected in series:
Let C1, C2, …… CN are the three capacitances connected in series and let V1, V2 ……VN are the potential
differences across the three capacitors. Let vs is the applied voltage across the combination and Ceq, the
combined or equivalent capacitance. For a series circuit, charge (current flow) on all capacitors is same
but potential difference across each is different.

Figure: (a) A circuit containing N capacitors in series. (b) The desired equivalent circuit
V=V1+V2+V3.
𝑄 𝑄 𝑄 𝑄
= + +….+
𝐶𝑒𝑞 𝐶1 𝐶2 𝐶𝑁

Therefore, if N capacitors are in series, then combined capacitance is given by

1 1 1 1
= + +….+
𝐶𝑒𝑞 𝐶1 𝐶2 𝐶𝑁

f) Total or equivalent inductance if N capacitors are connected in parallel:

Applying KCL to the original circuit,

is = i1 + i2 + …………. iN
𝑑𝑣 𝑑𝑣 𝑑𝑣
= C1 𝑑𝑡 + C2 𝑑𝑡 +………… …..+ CN 𝑑𝑡
𝑑𝑣
= (C1+ C2 + ……… CN ) 𝑑𝑡
𝑑𝑣
In equivalent circuit , is = Ceq. .
𝑑𝑡
Therefore, if N capacitors are in parallel, then combined capacitance is given by

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Worked example: Simplify the network using series-parallel combinations. Answer is:

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1.6.2 Star - Delta (Y- ∆) transformation

The methods of series, parallel and series – parallel combination of elements do not always lead to
simplification of networks. Such networks are handled by Star Delta transformation.

Figure a shows three resistances Ra, Rb, Rc connected in star to three nodes A, B, C and a common point N
& figure b shows three resistances connected in delta between the same three nodes A, B, C. If these two
networks are to be equivalent, then the resistance between any pair of nodes of the delta connected network
of a) must be the same as that between the same pair of nodes of the star – connected network of fig b).

Star resistances in terms of delta

Equating resistance between node pair AB

𝑅𝑎𝑏 𝑅𝑏𝑐 +𝑅𝑎𝑏 𝑅𝑐𝑎
Ra + Rb = Rab // (Rbc + Rca ) = _ (1)
𝑅𝑎𝑏 +𝑅𝑏𝑐 +𝑅𝑐𝑎

Similarly, for node pair BC

𝑅𝑏𝑐 𝑅𝑐𝑎 +𝑅𝑏𝑐 𝑅𝑎𝑏
Rb + Rc = Rbc // (Rca + Rab ) = _ (2)
𝑅𝑎𝑏 +𝑅𝑏𝑐 +𝑅𝑐𝑎
For Node pair CA
𝑅𝑐𝑎 𝑅𝑎𝑏 +𝑅𝑐𝑎 𝑅𝑏𝑐
Rc + Ra = Rca // (Rab + Rbc ) = _ (3)
𝑅𝑎𝑏 +𝑅𝑏𝑐 +𝑅𝑐𝑎

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 Subtracting (2) from (3) gives

𝑅𝑐𝑎 𝑅𝑎𝑏 +𝑅𝑏𝑐 𝑅𝑎𝑏

Ra – Rb = _ (4)
𝑅𝑎𝑏 +𝑅𝑏𝑐 +𝑅𝑐𝑎

Adding (1) and (4) gives

𝑅𝑎𝑏 𝑅𝑐𝑎
Ra = _ (5)
𝑅𝑎𝑏 +𝑅𝑏𝑐 +𝑅𝑐𝑎

Similarly

𝑅𝑏𝑐 𝑅𝑎𝑏
Rb = _ (6)
𝑅𝑎𝑏 +𝑅𝑏𝑐 +𝑅𝑐𝑎

𝑅𝑐𝑎 𝑅𝑏𝑐
Rc = _ (7)
𝑅𝑎𝑏 +𝑅𝑏𝑐 +𝑅𝑐𝑎

Thus the equivalent star resistance connected to a node is equal to the product of the two delta resistances
connected to the same node decided by the sum of delta resistances.

Delta resistances in terms of star resistances:

Similarly, by simplification of above equations, we get

𝑅𝑎 𝑅𝑏 𝑅𝑐 𝑅𝑎 𝑅𝑐 𝑅𝑎
Rab = Ra + Rb + Rbc = Rb + Rc + & Rca = Rc + Ra + and
𝑅𝑐 𝑅𝑏 𝑅𝑏

Thus the equivalent Delta resistance between two nodes is the sum of two star resistances connected to
those nodes plus the product of the same two star resistances divided by the third star resistance.

𝑅𝐴 𝑅𝐵 𝑅𝐴 𝑅𝐵
RAB = RA = RA + RB +
𝑅𝐴 +𝑅𝐵 +𝑅𝐶 𝑅𝐵

𝑅𝐵 𝑅𝐶 𝑅𝐴 𝑅𝐶
RCA = RB = RA + RC +
𝑅𝐴 +𝑅𝐵 +𝑅𝐶 𝑅𝐵

𝑅𝐶 𝑅𝐴 𝑅𝐵 𝑅𝐶
RCA = RC = RB + RC +
𝑅𝐴 +𝑅𝐵 +𝑅𝐶 𝑅𝐴

𝑹𝟐 𝑹
If all are similar equal to R RAB = = 𝑅𝐴 = 3𝑅
𝟑𝑹 𝟑

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1.8 (a) Voltage division rule: Voltage division is used to express the voltage across one of several series
resistors in terms of the voltage across the combination.

For the above network, the voltage across R2 is found via KVL and Ohm’s law:

Therefore

Or we can write that

Similarly, voltage across R1 can be found from

with the series combination of R1, R2, . . ., RN, then we have the general result for voltage division across a
string of N series resistors,

which allows us to compute the voltage v k that appears across an arbitrary resistor Rk of the series.

1.8 (b) Current division rule: Current division is used to express the current flowing in one of several parallel
resistors in terms of the currents combination in all resistors. For the following network, the current through
R2 is found via KCL and Ohm’s law:

Figure: An illustration of current division.

Current flowing through R2 resistor is given by

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And also for a parallel combination of N resistors, the current through resistor Rk is given by

Written in terms of conductances,

1.9 Source transformation technique:

A practical voltage source will have its source resistance in series with it, as mentioned in the below diagram
(a) RS is the source resistance of the voltage source VS.
Similarly, a practical current source will have its source resistance in parallel with it, as mentioned in the
below diagram (b) RP is the source resistance of the current source iS.

Figure: (a) A given practical voltage source connected to a load RL. (b) The equivalent practical current
source connected to the same load.

A simple calculation shows that the voltage across the load RL of Fig.(a) is

A similar calculation shows that the voltage across the load RL in Fig. b is

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The two practical sources are electrically equivalent, then, if

&
1.10 Nodal Analysis: (It is the simultaneous application of KCL & Ohm’s law)
In writing nodal equations, we perform the following steps:

1. Identify the number of nodes of the given network

2. Select a node as a reference node and also Assign voltages V1, V2, ……. V(N-1) to remaining (N-1) nodes.
The voltages are referenced with respect to reference node.
3. Apply KCL at each of (N-1) non-reference nodes and use Ohm’s law to express branch currents in
terms of voltages.
4. Solve the resulting simultaneous equations to obtain unknown node voltages (i.e. V1, V2, …………………….….
V(N-1).)

Example: Solve the circuit shown below by nodal analysis & determine the node voltages of all nodes

Solution: Assume that the currents are moving away from the nodes.
Applying KCL at Node 1,

Applying KCL at Node 2,

On solving the equation (1) and (2), we get V1=2V and V2=2V.

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1.11 Mesh Analysis: (It is the simultaneous application of KVL & Ohm’s law)

In writing nodal equations, we perform the following steps:

1. Identify the number of meshes for the given network

2. Assign mesh currents i1, i2, ………….. in to ‘n’ number of meshes
3. Apply KVL to each of n meshes & use Ohm’s law to express the voltages in terms of mesh currents.
4. Solve the resulting simultaneous equations to mesh currents (i.e. i1, i2, ………….. in.)

Example: Find the current through the 5Ω resistor is shown below.

Solution: Step 1: As there are 3 loops/mesh connections, three currents are required to analyse the network.
Step 2: Assign currents (clock-wise) to each loop as shown below

Step 3: Applying KVL to Mesh 1,

Applying KVL to Mesh 2,

Applying KVL to Mesh 3,

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Step 4: On solving the above 3 equations, we get

As 5Ω resistors is only connected to loop 2, hence current though it is I5Ω = I2 = 0.78A

What is the difference between mesh and loop?

Ans: A loop is any closed path through a circuit where no node is encountered more than once whereas
mesh is a closed path in a circuit that contains no other paths.

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1.12 SUPERNODE ANALYSIS
If there a present of voltage source along between two nodes, then we have to consider those two nodes as
a super node and we have to apply KCL & Ohm’s law in determining the node voltages. And the other voltage
equation is equal to difference of those two node voltages is equal to voltage source value which is in between
two nodes (or super node), measure in voltage source direction.
Example: Determine the current in the 5 W resistor for the network shown below.

Solution: Assume that the currents are moving away from the nodes. Applying KCL at Node 1,

Nodes 2 and 3 will form a supernode.

Writing voltage equation for the supernode, V2 – V3 = 20V

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Solving Eqs (i), (ii) and (iii), We get

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1.13 SUPERMESH ANALYSIS

If there a presence of current source or (current source in series with a resistance) between two meshes,
then, it is not possible to find the voltage across current source, so we have to consider those two meshes
as a supermesh and we have to apply KVL + Ohm’s law. The other current equation will be difference of
those mesh currents equal to current source value (measured in current source direction).
Example: Find the current through the 10 Ω resistor of the network shown below.

Solution: Applying KVL to Mesh 1, we get

Since meshes 2 and 3 contain a current source of 4 A, these two meshes will form a supermesh. A supermesh
is formed by two adjacent meshes that have a common current source. The direction of the current source
of 4 A and current (I3 – I2) are same, i.e., in the upward direction. Writing current equation to the supermesh,1

Applying KVL to the outer path of the supermesh

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Assignment (Theory) Questions:
1. State and explain K laws with an example.
2. Derive the expression for equivalent resistance/inductance/capacitance if “n” number of
resistors/inductors/capacitors are connected in (i) series (ii) parallel.
3. Distinguish between ideal and practical electrical sources.
4. Derive the relationship between star and delta connected resistive networks.

Assignment on Network Reduction Techniques:

1. Find the equivalent resistance between terminals A and B in the network shown below.

2. Find the equivalent resistance between terminals A and B in the network shown below.

Assignment on Nodal Analysis

1. List out the steps involved in analyzing a network using Nodal analysis.
2. Obtain the node voltages VA and VB for the network shown below.

3. Obtain the node voltages in the circuit [Ans: V1 = -6V and V2 = -42V ]

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Assignment on Mesh Analysis
1. List out the steps involved in analyzing a network using Mesh analysis.
2. Calculate the mesh currents for network shown below.

3. Find the voltage drop across the 5 Ω resistor for the network given below

Assignment on Super Node Analysis

1. Find the node voltages in the network shown [V1=10V, V2=20V, V3=30V and V4=40V]

Assignment on Super Mesh Analysis

1. Find the current through 3 Ω resistor for the network shown below.

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 2. Determine the power delivered by the voltage source and the current in the 10Ω resistor of the network
shown below

Key Outcomes: At the end of the course, student will be able to

☐ Understand and remember the various electrical elements definitions & properties
☐ Understand and apply Ohm’s law and Kirchhoff’s laws in analysing the various electrical circuits
☐ Solve complex circuits using mesh and nodal analysis techniques
Note: Students are advised to self-check the above outcomes achieved by individual

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