S V ENGINEERING COLLEGE KARAKAMBADI ROAD, TIRUPATI-517507

DEPARTMENT OF ELECTRICAL AND ELECTRONICS

ENGINEERING

LECTURE NOTES

19A02301T BASIC ELECTRICAL CIRCUITS

Regulation – R19

Academic Year (2020 – 21)

Year / Semester: II / I

CONTENTS

Unit- 1 Introduction to Electrical & Magnetic Circuits Page NO

1.1 Active Components 1

1.2 Passive Components 1

1.3 Source Transformation

1

1.4 Conversion of Voltage Source into Current Source

3 1.5 Kirchhoff’s Current Law and Kirchhoff’s Voltage Law 4

1.6 Star to Delta and Delta to Star Conversion

7

1.7 Faraday’s Law of Electromagnetic Induction

9

1.8 Composite Magnetic Circuit

12

1.9 Difference Between Electric Circuit and Magnetic Circuit:

13

1.10 Parallel Magnetic Circuit 16

Unit- II Single Phase A.C Circuits

2.1 AC Waveform Characteristics 18

2.2 Types of Periodic Waveform 19

2.3 Relationship Between Frequency and Periodic Time 20

2.4 The Average Value of an AC Waveform 21

2.5 The RMS Value of an AC Waveform 22

2.6 Form Factor and Crest Factor 22

2.7 Phase Difference 24

2.8 Complex Numbers and Phasors 25

2.9 Series RLC Circuit 27

2.10 Phasor Diagram for a Series RLC Circuit 29

2.11 Parallel RLC Circuit 31

Unit- III Three Phase A.C. Circuits

3.1 Three Phase System 35

3.2 3.2 Analysis of Balanced 3 Phase Circuit 36

3.3 3.3 Relation Between Phase Current and Line Current in Delta

Connection

39

3.4 3.4 Star Connection in a 3 Phase System 40

3.5 3.5 Relation Between Phase Voltage and Line Voltage in Star

Connection

41

3.6 3.6 Relation Between Phase Current and Line Current in Star

Connection

43

3.7 3.7 Advantages of 3 Phase Over Single Phase System 43

Unit- IV Network Theorems

4.1 Superposition Theorem 45

4.2 Steps for Solving network by Superposition Theorem 47

4.3 Reciprocity Theorem 49

4.4 Thevenin’s Theorem 50

4.5 Norton’s Theorem 53

4.6 Maximum Power Transfer Theorem 56

4.7 Millman’s Theorem 59

4.8 Tellegen’s Theorem 63

4.9 Compensation Theorem

66

Unit- V Network Topology

5.1 Graph 69

5.2 Types of Graphs 70

5.3 Directed Graph 71

5.4 Subgraph and its Types 72

5.5 Voltage Sources 74

5.6 Voltage Source in Series 77

5.7 Practical Voltage Source 78

5.8 Practical Voltage Source Characteristics 79

5.9 Dependent Voltage Source 81

5.10 Principle of Duality:

83

19A02301T BASIC ELECTRICAL CIRCUITS

Unit- 1 Introduction to Electrical & Magnetic Circuits

Electrical Circuits: Circuit Concept – Types of elements - Source Transformation-Voltage -

Current Relationship for Passive Elements. Kirchhoff’s Laws – Network Reduction

Techniques Series, Parallel, Series Parallel, Star-to-Delta or Delta-to-Star Transformation.

Examples Magnetic Circuits: Faraday’s Laws of Electromagnetic Induction-Concept of Self

and Mutual Inductance-Dot Convention-Coefficient of Coupling-Composite Magnetic Circuit-

Analysis of Series and Parallel Magnetic Circuits, MMF Calculations.

Unit- II Single Phase A.C Circuits

R.M.S, Average Values and Form Factor for Different Periodic Wave Forms – Sinusoidal

Alternating Quantities – Phase and Phase Difference – Complex and Polar Forms of

Representations, j-Notation, Steady State Analysis of R, L and C (In Series, Parallel and Series

Parallel Combinations) with Sinusoidal Excitation- Phasor diagrams - Concept of Power Factor

Concept of Reactance, Impedance, Susceptance and Admittance-Apparent Power, Active and

Reactive Power, Examples.

Unit- III Three Phase A.C. Circuits

Introduction - Analysis of Balanced Three Phase Circuits – Phase Sequence- Star and Delta

Connection - Relation between Line and Phase Voltages and Currents in Balanced Systems -

Measurement of Active and Reactive Power in Balanced and Unbalanced Three Phase

Systems. Analysis of Three Phase Unbalanced Circuits - Loop Method - Star Delta

Transformation Technique – for balanced and unbalanced circuits - Measurement of Active

and reactive Power –Advantages of Three Phase System.

Unit- IV Network Theorems

Superposition, Reciprocity, Thevenin’s, Norton’s, Maximum Power Transfer, Millmann’s,

Tellegen’s, and Compensation Theorems for D.C and Sinusoidal Excitations.

Unit- V Network Topology

Definitions – Graph – Tree, Basic Cutset and Basic Tieset Matrices for Planar Networks –

Loop and Nodal Methods of Analysis of Networks & Independent Voltage and Current Sources

–Duality & Dual Networks. Nodal Analysis, Mesh Analysis.

Text Books:

1. Fundamentals of Electric Circuits Charles K. Alexander and Matthew. N. O. Sadiku, Mc

Graw Hill, 5th Edition, 2013.

2. Circuit Theory (Analysis & Synthesis) A. Chakrabarti, Dhanpat Rai & Sons, 7th Revised

Edition, 2018.

9 Page

Reference Books:

1. Engineering circuit analysis William Hayt and Jack E. Kemmerly, Mc Graw Hill

Company, 7th Edition, 2006.

2. Network Analysis M.E Van Valkenberg, Prentice Hall (India), 3rd Edition, 1999.

3. Electrical Engineering Fundamentals V. Del Toro, Prentice Hall International, 2nd

Edition, 2019.

4. Electric Circuits- Schaum’s Series, Mc Graw Hill, 5th Edition, 2010.

5. Electrical Circuit Theory and Technology John Bird, Routledge, Taylor & Francis, 5th

Edition, 2014.

1

Unit- 1 Introduction to Electrical & Magnetic Circuits

1.1Active Components:

Those devices or components which required external source to their operation is called Active

Components.

For Example: Diode, Transistors, SCR etc…

Explanation and Example: As we know that Diode is an Active Components. So it is required

an External Source to its operation.

Because, If we connect a Diode in a Circuit and then connect this circuit to the Supply voltage.,

then Diode will not conduct the current Until the supply voltage reach to 0.3(In case of

Germanium) or 0.7V(In case of Silicon).

1.2 Passive Components:

Those devices or components which do not required external source to their operation is called

Passive Components.

For Example: Resistor, Capacitor, Inductor etc…

Explanation and Example: Passive Components do not require external source to their

operation.

Like a Diode, Resistor does not require 0.3 0r 0.7 V. I.e., when we connect a resistor to the

supply voltage, it starts work automatically without using a specific voltage

1.3 Source Transformation

It means replacing one source by an equivalent source. A practical voltage source can be

transformed into an equivalent practical current source and similarly a practical current source

into voltage source.

Any practical voltage source or simply a voltage source consists of an ideal voltage source in

series with an internal resistance or impedance (for an ideal source this impedance will be zero),

the output voltage becomes independent of the load current. Cells, batteries and generators are

the example of the voltage source.

For any practical current source or simply current source, there is an ideal current source in

parallel with the internal resistance or impedance, for ideal current source this parallel

impedance is infinity.

The semiconductor devices like transistors, etc. are treated as a current source or an output

produce by the direct or alternating voltage source is called direct and alternating current

source, respectively.

The voltage and current source are mutually transferable or in other words the source

transformation i.e. voltage to the current source and current to a voltage source can be done.

Let us understand this by considering a circuit given below:

2

Figure A represents a practical voltage source in series with the internal resistance rv, while

figure B represents a practical current source with parallel internal resistance ri

For the practical voltage source the load current will be given by the equation:

Where,

iLv is the load current for the practical voltage source

V is the voltage

rv is the internal resistance of the voltage source

rL is the load resistance

It is assumed that the load resistance rL is connected at the terminal x-y. Similarly for the

practical current source, the load current is given as:

Where,

iLi is the load current for the practical current source

I is the current

ri is the internal resistance of the current source

rL is the load resistance connected across the terminal x-y in the figure B

Two sources become identical, when we will equate equation (1) and equation (2)

3

However, for the current source, the terminal voltage at x-y would be Iri, x-y terminal are

open. i.e.

V = I x ri

Therefore, we will get,

Therefore, for any practical voltage source, if the ideal voltage is V and internal resistance be

rv, the voltage source can be replaced by a current source I with the internal resistance in

parallel with the current source.

Source Transformation:

1.4Conversion of Voltage Source into Current Source

When the voltage source is connected with the resistance in series and it has to be converted

into the current source than the resistance is connected in parallel with the current source as

shown in the above figure.

Where Is = Vs /R

Conversion of Current Source into Voltage Source

In the above circuit diagram a current source which is connected in parallel with the

resistance is transformed into a voltage source by placing the resistance in series with the

voltage source.

Where, Vs = Is / R

4

1.5Kirchhoff’s Current Law and Kirchhoff’s Voltage Law

Kirchhoff’s Law: A German physicist Gustav Kirchhoff developed two laws enabling easy

analysis of interconnection of any number of circuit elements. The first law deals with the flow

of current and is popularly known as Kirchhoff’s Current Law (KCL) while the second one

deals with the voltage drop in a closed network and is known as Kirchhoff’s Voltage

Law (KVL).

The KCL states that the summation of current at a junction remains zero and according to KVL

the sum of the electromotive force and the voltage drops in a closed circuit remains zero.

While applying the KCL the incoming current is taken as positive and the outgoing current is

taken as negative. Similarly, While applying KVL, the rise in potential is taken as positive and

the fall in potential is taken as negative.

The KVL and KCL help in finding the analogous electrical resistance and impedances of the

complex system. It also determines the current flowing through each branch of the network.

Kirchhoff’s Current Law

Kirchhoff’s Current Law states that” the algebraic sum of all the currents at any node point or

a junction of a circuit is zero”.

Σ I = 0

Considering the above figure as per the Kirchhoff’s Current Law:

i1 + i2 – i3 – i4 – i5 + i6 = 0 ……… (1)

The direction of incoming currents to a node is taken as positive while the outgoing currents

are taken as negative. The reverse of this can also be taken, i.e. incoming current as negative

or outgoing as positive. It depends upon your choice.

The equation (1) can also be written as:

i1 + i2 + i6 = i3 + i4 + i5

Sum of incoming currents = Sum of outgoing currents

5

According to the Kirchhoff’s Current Law, the algebraic sum of the currents entering a node

must be equal to the algebraic sum of the currents leaving the node in an electrical network.

Kirchhoff’s Voltage Law

Kirchhoff’s Voltage Law states that the algebraic sum of the voltages (or voltage drops) in

any closed path of a network that is transverse in a single direction is zero. In other words, in

a closed circuit, the algebraic sum of all the EMFs and the algebraic sum of all the voltage

drops (product of current (I) and resistance (R)) is zero.

Σ E + Σ V = 0

The above figure shows closed-circuit also termed as a mesh. As per the Kirchhoff’s Voltage

Law:

Here, the assumed current I causes a positive voltage drop when flowing from the positive to

negative potential while negative potential drop when the current flowing from negative to

the. positive potential

Considering the other figure shown below and assuming the direction of the current i

6

It is seen that the voltage V1 is negative in both the equation (2) and equation (3) while V2 is

negative in the equation (2) but positive in the equation (3). This is because of the change in

the direction of the current assumed in both the figures.

In figure A, the current in both the source V1 and V2 flows from negative to positive polarity

while in figure B the current in the source V1 is negative to positive but for V2 is positive to

negative polarity.

For the dependent sources in the circuit, KVL can also be applied. In case of the calculation

of the power of any source, when the current enters the source, the power is absorbed by the

sources while the source delivers the power if the current is coming out of the source.

It is important to know some of the terms used in the circuit while applying KCL and KVL

like node, Junction, branch, loop, mesh. They are explained with the help of a circuit shown

below:

7

Node: A node is a point in the network or circuit where two or more circuit elements are

joined. For example, in the above circuit diagram, A and B is the node points.

Junction: A junction is a point in the network where three or more circuit elements are joined.

It is a point where the current is divided. In the above circuit, B and D are the junctions.

Branch :The part of a network, which lies between the two junction points is called a Branch.

In the above circuit DAB, BCD and BD are the branches of the circuit.

Loop: A closed path of a network is called a loop. ABDA, BCDB are loops in the above circuit

diagram shown.

Mesh: The most elementary form of a loop which cannot be further divided is called a mesh.

1.6 Star to Delta and Delta to Star Conversion

The conversion or transformation or replacement of the star connected load network to a Delta

connected network and similarly, a delta connected network to a star network is done by Star

to Delta or Delta to Star Conversion.

Star to Delta Conversion

In star to delta conversion, the star-connected load is to be converted into delta connection.

Suppose we have a star connected load as shown in the figure above, and it has to be

converted into a delta connection as shown in figure B.

The following Delta values are as follows:

8

Hence, if the values of ZA, ZB and ZC are known, therefore by knowing these values and by

putting them in the above equations, you can convert a star connection into a delta

connection.

Delta to Star Conversion

Similarly, a delta connection network is given as shown above, in figure B and it has to be

transformed into a star connection, as shown above, in the figure A. The following formulas

given below are used for the conversion:

If the values of Z1, Z2 and Z3 are given, then by putting these values of the Impedances in

the above equations, the conversion of delta connection into star connection can be

performed. As Impedance (Z) is the vector quantity, therefore all the calculations are done in

Polar and Rectangular form.

Steps for Analysis of 3 phase unbalanced Delta Connected Load

Let us take an example of Delta connected unbalanced load connected to 400 V, 3 phase

supply as shown in the figure below:

9

The following steps are given below to solve the 3 phase unbalanced delta connected loads.

Step 1 – Solve, each phase current I1, I2 and I3 as in the single-phase circuit.

As

Step 2 – Compute the line currents IL1, IL2 and IL3, in rectangular form.

Step 3 – Now, compute the power of different phases.

Step 4 – Calculate the total power by the equation shown below.

A similar procedure will be followed for solving the unbalanced Star connected load. Firstly,

the star impedance will be converted into the corresponding delta impedance. The remaining

calculations will be followed as for the Delta connected load as shown above.

1.7 Faraday’s Law of Electromagnetic Induction

10

The name Faraday’s Law of Electromagnetic Induction is given in the name of a famous

scientist Michael Faraday in the 1930’s. It gives the relationship between electric voltage and

changing magnetic field. Faraday’s Law of Electromagnetic Induction states that “ the

magnitude of voltage is directly proportional to the rate of change of flux.” that means the

voltage is induced in the circuit when there is relative motion between a magnetic field and the

conductor.

Electromagnetic Induction

In a closed circuit when the current flows and the emf is induced, therefore the phenomenon

by which an emf is induced in a circuit when magnetic flux linking with it changes is

called Electro Magnetic Induction.

This can be explained by taking an example

Consider a coil having a large number of turns to which the galvanometer is connected

Case 1: – When the coil is stationary, and the magnet is moving

When a permanent bar magnet is taken nearer to the coil (position 2) or away from the coil

(position 1) as shown in the above figure, the deflection takes place in the galvanometer. The

deflections are the opposite in both cases.

Case 2: – When the coil is moving, and the magnet is stationary.

11

If the bar magnet is kept stationary and the coil is brought nearer to the magnet(position 1) or

away from the magnet (position 2), the deflection will take place in the galvanometer.

In both cases, the direction of the needle will be opposite. This can be explained as suppose if

the magnet is brought nearer than the needle will deflect towards the right and if moved away

from the magnet it shows deflection on the left side.

Case 3: – When the magnet and the coil both are stationary

When both the magnet and the coil is kept stationary, there will be no deflection in the coil

regardless of how much flux is linked with the coil

Following points are analysed

The deflection of the galvanometer needle indicates that the emf is induced in the coil. The

deflection takes place only when the flux linking with the circuit changes. i.e either the magnet

or the coil is in motion.

The direction of the induced emf in the coil depends upon the direction of the magnetic field

and the direction of motion of the coil.

Coefficient of Coupling:

The amount of coupling between the inductively coupled coils is expressed in terms of the

coefficient of coupling, which is defined as

where

M = mutual inductance between the coils

L1 = self inductance of the first coil, and

L2 = self inductance of the second coil

Coefficient of coupling is always less than unity, and has a maximum value of 1 (or 100%).

This case, for which K = 1, is called perfect coupling, when the entire flux of one coil links the

other. The greater the coefficient of coupling between the two coils, the greater the mutual

inductance between them, and vice-versa. It can be expressed as the fraction of the magnetic

flux produced by the current in one coil that links the other coil.

12

1.8 COMPOSITE MAGNETIC CIRCUIT

Consider a toroid composed of three different magnetic materials of different permeabilities,

areas and lengths excited by a coil of N turns.

With a current of I amperes as shown in Fig. (a). The lengths of sections AB, BC and CA are

l1,l2 and l3 respectively. Each section will have its own reluctance and permeability. Since

all of them are joined in series, the total reluctance of the combined magnetic circuit is given

by

Reluctance of a Composite Magnetic Circuit

Consider a composite magnetic circuit as shown in Fig. (a), which consists of three types of

the magnetic materials A, B and C with l1,l2 and l3 their mean lengths, A1, A2 and A3 their

cross-sectional area and μ1,μ2 and μ3 their

13

FIGURE (a) A composite magnetic circuit having two iron specimens A and B

Equation (1) suggests that the reluctances of different materials carried by a single magnetic

line of force add up to each other, which is analogous to the addition of resistance connected

in series in an electric circuit.

1.9 Difference Between Electric Circuit and Magnetic Circuit:

A Difference Between Electric Circuit and Magnetic Circuit are shown in Figs. (a) and (b)

respectively.

14

Figure (a) represents an electric circuit with three resistances connected in series, the dc source

E drives the current I through all the three resistances whose voltage drops are V1,V2 and V3.

Hence, E = V1 + V2 + V3, also E = I(R1 + R2 + R3). We also know that R = ρl/a, where ρ is

the specific resistance of the material, l is the length of the wire of the resistive material and a

is the area of cross-section of the wire.

The drop across each resistor V = RI =ρl I/a

or

Voltage drop per unit length = specific resistance x current density.

Let us consider the magnetic circuit in Fig. (b). The MMF of the circuit is given by ζ = NI,

drives the flux Φ around the three parts of the circuit which are in series. Each part has a

reluctance R = 1/μ . l/a, where l is the length and a is the area of cross-section of each arm. The

mmf of the magnetic circuit is given by ζ = ζ1 + ζ2 + ζ3 . ζ = Φ(R1 + R2 + R3) where

R1 R2 and R3 are the reluctances of the portion 1, 2 and 3 respectively.

Also

1/μ can be termed as reluctance of a cubic metre of magnetic material from

which, the above equation gives the mmf per unit length (intensity) which is analogous to the

voltage per unit length. Parallels Difference Between Electric Circuit and Magnetic Circuit

quantities are shown in Table.

15

Thus, it is seen that the magnetic reluctance is analogous to resistance, mmf is analogous to

emf and flux is analogous to current. These analogies are useful in magnetic circuit

calculations. Though we can draw many parallels between the two circuits, the following

differences do exists.

The electric current is a true flow but there is no flow in a magnetic flux. For a given

temperature, ρ is independent of the strength of the current, but μ is not independent of the flux.

In an electric circuit energy is expended so long as the current flows, but in a magnetic circuit

energy is expended only in creating the flux, and not in maintaining it. Parallels between the

quantities are shown in Table

Series Magnetic Circuit:

A series magnetic circuit is analogous to a series electric circuit. Kirchhoff s laws are applicable

to magnetic circuits also. Consider a ring specimen having a magnetic path of l meters, area of

cross-section (A)m2 with a mean radius of R meters having a coil of N turns carrying I amperes

wound uniformly as shown in Fig.. MMF is responsible for the establishment of flux in the

magnetic medium. This mint acts along the magnetic lines of force. The flux produced by the

circuit is given by

16

The magnetic field intensity of the ring is given by

Where l is the mean length of the magnetic path and is given by 2πR.

Flux density

NI is the mint of the magnetic circuit, which is analogous to emf in electric circuit. l/μ0μrA is

the reluctance of the Series Magnetic Circuit which is analogous to resistance in electric

circuit.

1.10 Parallel Magnetic Circuit:

We have seen that a series magnetic circuit carries the same flux and the total mmf required

to produce a given quantity of flux is the sum of the mmf’ s for the separate parts. In a

Parallel Magnetic Circuit, different parts of the circuit are in parallel. For such circuits the

Kirchhoff s laws, in their analogous magnetic form can be applied for the analysis. Consider

an iron core having three limbs A, B and C as shown in Fig.

17

A Coil with N turns is arranged around limb A which carries a current I amperes. The flux

produced by the coil in limb A. ΦA is divided between limbs B and C and each equal to ΦA/2.

The reluctance offered by the two parallel paths is equal to the half the reluctance of each path

(Assuming equal lengths and cross sectional areas). Similar to ICirchhoff s current law in an

electric circuit, the total magnetic flux directed towards a junction in a magnetic circuit is equal

to the sum of the magnetic fluxes directed away from that junction.

Accordingly ΦA = ΦB + ΦC or ΦA – ΦB – ΦC = 0. The electrical equivalent of the above

circuit is shown in Fig. 10.31(b). Similar to Kirchhoff s second law, in a closed magnetic

circuit, the resultant mmf is equal to the algebraic sum of the products of field strength and the

length of each part in the closed path. Thus applying the law to the first loop in Fig. (a), we get

The mmf across the two parallel paths is identical.

Therefore NI is also equal to

18

Unit- II Single Phase A.C Circuits

An alternating function or AC Waveform on the other hand is defined as one that varies in both

magnitude and direction in more or less an even manner with respect to time making it a “Bi-

directional” waveform. An AC function can represent either a power source or a signal source

with the shape of an AC waveform generally following that of a mathematical sinusoid being

defined as: A(t) = Amax*sin(2πƒt).

The term AC or to give it its full description of Alternating Current, generally refers to a time-

varying waveform with the most common of all being called a Sinusoid better known as

a Sinusoidal Waveform. Sinusoidal waveforms are more generally called by their short

description as Sine Waves. Sine waves are by far one of the most important types of AC

waveform used in electrical engineering.

The shape obtained by plotting the instantaneous ordinate values of either voltage or current

against time is called an AC Waveform. An AC waveform is constantly changing its polarity

every half cycle alternating between a positive maximum value and a negative maximum value

respectively with regards to time with a common example of this being the domestic mains

voltage supply we use in our homes.

This means then that the AC Waveform is a “time-dependent signal” with the most common

type of time-dependant signal being that of the Periodic Waveform. The periodic or AC

waveform is the resulting product of a rotating electrical generator. Generally, the shape of any

periodic waveform can be generated using a fundamental frequency and superimposing it with

harmonic signals of varying frequencies and amplitudes but that’s for another tutorial.

Alternating voltages and currents can not be stored in batteries or cells like direct current (DC)

can, it is much easier and cheaper to generate these quantities using alternators or waveform

generators when they are needed. The type and shape of an AC waveform depends upon the

generator or device producing them, but all AC waveforms consist of a zero voltage line that

divides the waveform into two symmetrical halves.

2.1 AC Waveform Characteristics

• The Period, (T) is the length of time in seconds that the waveform takes to repeat itself from

start to finish. This can also be called the Periodic Time of the waveform for sine waves, or

the Pulse Width for square waves.

• The Frequency, (ƒ) is the number of times the waveform repeats itself within a one second

time period. Frequency is the reciprocal of the time period, ( ƒ = 1/T ) with the unit of

frequency being the Hertz, (Hz).

• The Amplitude (A) is the magnitude or intensity of the signal waveform measured in volts or

amps.

In our tutorial about Waveforms ,we looked at different types of waveforms and said that

“Waveforms are basically a visual representation of the variation of a voltage or current plotted

to a base of time”. Generally, for AC waveforms this horizontal base line represents a zero

condition of either voltage or current. Any part of an AC type waveform which lies above the

horizontal zero axis represents a voltage or current flowing in one direction.

19

Likewise, any part of the waveform which lies below the horizontal zero axis represents a

voltage or current flowing in the opposite direction to the first. Generally for sinusoidal AC

waveforms the shape of the waveform above the zero axis is the same as the shape below it.

However, for most non-power AC signals including audio waveforms this is not always the

case.

The most common periodic signal waveforms that are used in Electrical and Electronic

Engineering are the Sinusoidal Waveforms. However, an alternating AC waveform may not

always take the shape of a smooth shape based around the trigonometric sine or cosine function.

AC waveforms can also take the shape of either Complex Waves, Square Waves or Triangular

Waves and these are shown below..

2.2Types of Periodic Waveform

The time taken for an AC Waveform to complete one full pattern from its positive half to its

negative half and back to its zero baseline again is called a Cycle and one complete cycle

contains both a positive half-cycle and a negative half-cycle. The time taken by the waveform

to complete one full cycle is called the Periodic Time of the waveform, and is given the symbol

“T”.

The number of complete cycles that are produced within one second (cycles/second) is called

the Frequency, symbol ƒ of the alternating waveform. Frequency is measured in Hertz, ( Hz )

named after the German physicist Heinrich Hertz.

Then we can see that a relationship exists between cycles (oscillations), periodic time and

frequency (cycles per second), so if there are ƒ number of cycles in one second, each individual

cycle must take 1/ƒ seconds to complete.

20

2.3 Relationship Between Frequency and Periodic Time

AC Waveform Example No1

1. What will be the periodic time of a 50Hz waveform and 2. what is the frequency of an AC

waveform that has a periodic time of 10mS.

1).

2).

Frequency used to be expressed in “cycles per second” abbreviated to “cps”, but today it is

more commonly specified in units called “Hertz”. For a domestic mains supply the frequency

will be either 50Hz or 60Hz depending upon the country and is fixed by the speed of rotation

of the generator. But one hertz is a very small unit so prefixes are used that denote the order of

magnitude of the waveform at higher frequencies such as kHz, MHz and even GHz.

Amplitude of an AC Waveform

As well as knowing either the periodic time or the frequency of the alternating quantity, another

important parameter of the AC waveform is Amplitude, better known as its Maximum or Peak

value represented by the terms, Vmax for voltage or Imax for current.

The peak value is the greatest value of either voltage or current that the waveform reaches

during each half cycle measured from the zero baseline. Unlike a DC voltage or current which

has a steady state that can be measured or calculated using Ohm’s Law, an alternating quantity

is constantly changing its value over time.

For pure sinusoidal waveforms this peak value will always be the same for both half cycles

( +Vm = -Vm ) but for non-sinusoidal or complex waveforms the maximum peak value can be

very different for each half cycle. Sometimes, alternating waveforms are given a peak-to-

peak, Vp-p value and this is simply the distance or the sum in voltage between the maximum

peak value, +Vmax and the minimum peak value, -Vmax during one complete cycle.

21

2.4 The Average Value of an AC Waveform

The average or mean value of a continuous DC voltage will always be equal to its maximum

peak value as a DC voltage is constant. This average value will only change if the duty cycle

of the DC voltage changes. In a pure sine wave if the average value is calculated over the full

cycle, the average value would be equal to zero as the positive and negative halves will cancel

each other out. So the average or mean value of an AC waveform is calculated or measured

over a half cycle only and this is shown below.

Average Value of a Non-sinusoidal Waveform

To find the average value of the waveform we need to calculate the area underneath the

waveform using the mid-ordinate rule, trapezoidal rule or the Simpson’s rule found commonly

in mathematics. The approximate area under any irregular waveform can easily be found by

simply using the mid-ordinate rule.

The zero axis base line is divided up into any number of equal parts and in our simple example

above this value was nine, ( V1 to V9 ). The more ordinate lines that are drawn the more

accurate will be the final average or mean value. The average value will be the addition of all

the instantaneous values added together and then divided by the total number. This is given as.

Average Value of an AC Waveform

Where: n equals the actual number of mid-ordinates used.

For a pure sinusoidal waveform this average or mean value will always be equal

to 0.637*Vmax and this relationship also holds true for average values of current.

22

2.5The RMS Value of an AC Waveform

The average value of an AC waveform that we calculated above as being: 0.637*Vmax is NOT

the same value we would use for a DC supply. This is because unlike a DC supply which is

constant and and of a fixed value, an AC waveform is constantly changing over time and has

no fixed value. Thus the equivalent value for an alternating current system that provides the

same amount of electrical power to a load as a DC equivalent circuit is called the “effective

value”.

The effective value of a sine wave produces the same I2*R heating effect in a load as we would

expect to see if the same load was fed by a constant DC supply. The effective value of a sine

wave is more commonly known as the Root Mean Squared or simply RMS value as it is

calculated as the square root of the mean (average) of the square of the voltage or current.

That is Vrms or Irms is given as the square root of the average of the sum of all the squared

mid-ordinate values of the sine wave. The RMS value for any AC waveform can be found from

the following modified average value formula as shown.

RMS Value of an AC Waveform

Where: n equals the number of mid-ordinates.

For a pure sinusoidal waveform this effective or R.M.S. value will always be equal

too: 1/√2*Vmax which is equal to 0.707*Vmax and this relationship holds true for RMS

values of current. The RMS value for a sinusoidal waveform is always greater than the average

value except for a rectangular waveform. In this case the heating effect remains constant so the

average and the RMS values will be the same.

One final comment about R.M.S. values. Most multimeters, either digital or analogue unless

otherwise stated only measure the R.M.S. values of voltage and current and not the average.

Therefore when using a multimeter on a direct current system the reading will be equal

to I = V/R and for an alternating current system the reading will be equal to Irms = Vrms/R.

Also, except for average power calculations, when calculating RMS or peak voltages, only use

VRMS to find IRMS values, or peak voltage, Vp to find peak current, Ip values. Do not mix

them together as Average, RMS or Peak values of a sine wave are completely different and

your results will definitely be incorrect.

2.6Form Factor and Crest Factor

Although little used these days, both Form Factor and Crest Factor can be used to give

information about the actual shape of the AC waveform. Form Factor is the ratio between the

average value and the RMS value and is given as.

23

For a pure sinusoidal waveform the Form Factor will always be equal to 1.11. Crest Factor is

the ratio between the R.M.S. value and the Peak value of the waveform and is given as.

For a pure sinusoidal waveform the Crest Factor will always be equal to 1.414.

Phase

Definition: The phase of an alternating quantity is defined as the divisional part of a cycle

through which the quantity moves forward from a selected origin. When the two quantities

have the same frequency, and their maximum and minimum point achieve at the same point,

then the quantities are said to have in the same phase.

Consider the two alternating currents Im1 and Im2 shown in the figure below. Both the quantity

attains their maximum and minimum peak point at the same time. And the zero value of both

the quantities establishes at the same instant

24

2.7 Phase Difference

Definition: The phase difference between the two electrical quantities is defined as the angular

phase difference between the maximum possible value of the two alternating quantities having

the same frequency.

In other words, the two alternating quantities have phase difference when they have the same

frequency, but they attain their zero value at the different instant. The angle between zero points

of two alternating quantities is called angle of phase differences.

Consider the two alternating currents of magnitudes Im1 and Im2 are shown vectorially.Both

the vector is rotating at the same angular velocity of ω radians per seconds. The two current

obtains the zero value at different instants. Therefore, they are said to have the phase difference

of angle φ.

The quantity which attains its +ve maximum value before the other is called a leading quantity,

whereas the quantity which reaches its maximum positive value after the other, is known as a

lagging quantity. The current Im1 is leading the current on Im2 or in other words, current

Im2 is the lagging current on Im1.

25

Cycle: When the alternating quantity goes from through a complete set of +ve or –ve value or

goes through 360° electrical degrees, then it is said to have completed a cycle completely.

2.8 Complex Numbers and Phasors

The mathematics used in Electrical Engineering to add together resistances, currents or DC

voltages use what are called “real numbers” used as either integers or as fractions.

But real numbers are not the only kind of numbers we need to use especially when dealing with

frequency dependent sinusoidal sources and vectors. As well as using normal or real

numbers, Complex Numbers were introduced to allow complex equations to be solved with

numbers that are the square roots of negative numbers, √-1.

In electrical engineering this type of number is called an “imaginary number” and to distinguish

an imaginary number from a real number the letter “ j ” known commonly in electrical

engineering as the j-operator, is used. Thus the letter “j” is placed in front of a real number to

signify its imaginary number operation.

Complex Numbers using the Rectangular Form

In the last tutorial about Phasors, we saw that a complex number is represented by a real part

and an imaginary part that takes the generalised form of:

26

Where:

Z - is the Complex Number representing the Vector

x - is the Real part or the Active component

y - is the Imaginary part or the Reactive component

j - is defined by √-1

In the rectangular form, a complex number can be represented as a point on a two dimensional

plane called the complex or s-plane. So for example, Z = 6 + j4 represents a single point whose

coordinates represent 6 on the horizontal real axis and 4 on the vertical imaginary axis as

shown.

Complex Numbers using the Complex or s-plane

Four Quadrant Argand Diagram

27

2.9 Series RLC Circuit

The series RLC circuit above has a single loop with the instantaneous current flowing through

the loop being the same for each circuit element. Since the inductive and capacitive

reactance’s XL and XC are a function of the supply frequency, the sinusoidal response of a

series RLC circuit will therefore vary with frequency, ƒ. Then the individual voltage drops

across each circuit element of R, L and C element will be “out-of-phase” with each other as

defined by:

i(t) = Imax sin(ωt)

The instantaneous voltage across a pure resistor, VR is “in-phase” with current

The instantaneous voltage across a pure inductor, VL “leads” the current by 90o

The instantaneous voltage across a pure capacitor, VC “lags” the current by 90o

Therefore, VL and VC are 180o “out-of-phase” and in opposition to each other.

For the series RLC circuit above, this can be shown as:

The amplitude of the source voltage across all three components in a series RLC circuit is made

up of the three individual component voltages, VR, VL and VC with the current common to all

28

three components. The vector diagrams will therefore have the current vector as their reference

with the three voltage vectors being plotted with respect to this reference as shown below.

Individual Voltage Vectors

This means then that we can not simply add together VR, VL and VC to find the supply

voltage, VS across all three components as all three voltage vectors point in different directions

with regards to the current vector. Therefore we will have to find the supply voltage, VS as

the Phasor Sum of the three component voltages combined together vectorially.

Kirchhoff’s voltage law ( KVL ) for both loop and nodal circuits states that around any closed

loop the sum of voltage drops around the loop equals the sum of the EMF’s. Then applying

this law to the these three voltages will give us the amplitude of the source voltage, VS as.

Instantaneous Voltages for a Series RLC Circuit

The phasor diagram for a series RLC circuit is produced by combining together the three

individual phasors above and adding these voltages vectorially. Since the current flowing

through the circuit is common to all three circuit elements we can use this as the reference

vector with the three voltage vectors drawn relative to this at their corresponding angles.

29

The resulting vector VS is obtained by adding together two of the vectors, VL and VC and then

adding this sum to the remaining vector VR. The resulting angle obtained

between VS and i will be the circuits phase angle as shown below

2.10 Phasor Diagram for a Series RLC Circuit

We can see from the phasor diagram on the right hand side above that the voltage vectors

produce a rectangular triangle, comprising of hypotenuse VS, horizontal axis VR and vertical

axis VL – VC Hopefully you will notice then, that this forms our old favourite the Voltage

Triangle and we can therefore use Pythagoras’s theorem on this voltage triangle to

mathematically obtain the value of VS as shown.

Please note that when using the above equation, the final reactive voltage must always be

positive in value, that is the smallest voltage must always be taken away from the largest

voltage we can not have a negative voltage added to VR so it is correct to

have VL – VC or VC – VL. The smallest value from the largest otherwise the calculation

of VS will be incorrect.

We know from above that the current has the same amplitude and phase in all the components

of a series RLC circuit. Then the voltage across each component can also be described

mathematically according to the current flowing through, and the voltage across each element

as.

30

By substituting these values into the Pythagoras equation above for the voltage triangle will

give us:

So we can see that the amplitude of the source voltage is proportional to the amplitude of the

current flowing through the circuit. This proportionality constant is called the Impedance of

the circuit which ultimately depends upon the resistance and the inductive and capacitive

reactance’s.

Then in the series RLC circuit above, it can be seen that the opposition to current flow is made

up of three components, XL, XC and R with the reactance, XT of any series RLC circuit being

defined as: XT = XL – XC or XT = XC – XL whichever is greater. Thus the total impedance

of the circuit being thought of as the voltage source required to drive a current through it.

The Impedance of a Series RLC Circuit

As the three vector voltages are out-of-phase with each other, XL, XC and R must also be “out-

of-phase” with each other with the relationship between R, XL and XC being the vector sum

of these three components. This will give us the RLC circuits overall impedance, Z. These

circuit impedance’s can be drawn and represented by an Impedance Triangle as shown below.

31

The Impedance Triangle for a Series RLC Circuit

The phase angle, θ between the source voltage, VS and the current, i is the same as for the

angle between Z and R in the impedance triangle. This phase angle may be positive or negative

in value depending on whether the source voltage leads or lags the circuit current and can be

calculated mathematically from the ohmic values of the impedance triangle as:

2.11 Parallel RLC Circuit

In the above parallel RLC circuit, we can see that the supply voltage, VS is common to all three

components whilst the supply current IS consists of three parts. The current flowing through

the resistor, IR, the current flowing through the inductor, IL and the current through the

capacitor, IC.

32

But the current flowing through each branch and therefore each component will be different to

each other and also to the supply current, IS. The total current drawn from the supply will not

be the mathematical sum of the three individual branch currents but their vector sum.

Like the series RLC circuit, we can solve this circuit using the phasor or vector method but this

time the vector diagram will have the voltage as its reference with the three current vectors

plotted with respect to the voltage. The phasor diagram for a parallel RLC circuit is produced

by combining together the three individual phasors for each component and adding the currents

vectorially.

Since the voltage across the circuit is common to all three circuit elements we can use this as

the reference vector with the three current vectors drawn relative to this at their corresponding

angles. The resulting vector current IS is obtained by adding together two of the

vectors, IL and IC and then adding this sum to the remaining vector IR. The resulting angle

obtained between V and IS will be the circuits phase angle as shown below

Phasor Diagram for a Parallel RLC Circuit

We can see from the phasor diagram on the right hand side above that the current vectors

produce a rectangular triangle, comprising of hypotenuse IS, horizontal axis IR and vertical

axis IL – IC Hopefully you will notice then, that this forms a Current Triangle. We can

therefore use Pythagoras’s theorem on this current triangle to mathematically obtain the

individual magnitudes of the branch currents along the x-axis and y-axis which will determine

the total supply current IS of these components as shown.

33

Impedance of a Parallel RLC Circuit

Admittance of a Parallel RLC Circuit

Admittance Triangle for a Parallel RLC Circuit

34

35

Unit- III Three Phase A.C. Circuits

3.1 Three Phase System

Definition: The system which has three phases, i.e., the current will pass through the three

wires, and there will be one neutral wire for passing the fault current to the earth is known as

the three phase system. In other words, the system which uses three wires for generation,

transmission and distribution is known as the three phase system. The three phase system is

also used as a single phase system if one of their phase and the neutral wire is taken out from

it. The sum of the line currents in the 3-phase system is equal to zero, and their phases are

differentiated at an angle of 120º

The three-phase system has four wire, i.e., the three current carrying conductors and the one

neutral. The cross section area of the neutral conductor is half of the live wire. The current in

the neutral wire is equal to the sum of the line current of the three wires and consequently equal

to √3 times the zero phase sequence components of current.

The three-phase system has several advantages like it requires fewer conductors as compared

to the single phase system. It also gives the continuous supply to the load. The three-phase

system has higher efficiency and minimum losses.

The three phase system induces in the generator which gives the three phase voltage of equal

magnitude and frequency. It provides an uninterruptible power, i.e., if one phase of the system

is disturbed, then the remaining two phases of the system continue supplies the power.The

magnitude of the current in one phase is equal to the sum of the current in the other two phases

of the system.

Circuit Analysis of 3 Phase System – Balanced Condition

The electrical system is of two types i.e., the single-phase system and the three-phase system.

The single-phase system has only one phase wire and one return wire thus it is used for low

power transmission.

The three-phase system has three live wire and one returns the path. The three-phase system is

used for transmitting a large amount of power. The 3 phase system is divided mainly into two

types. One is a balanced three-phase system and another one is an unbalanced three-phase

system.

36

The balance system is one in which the load are equally distributed in all the three phases of

the system. The magnitude of voltage remains same in all the three phases and it is separated

by an angle of 120º.

In the unbalance system the magnitude of voltage in all the three phases becomes different.

3.2 Analysis of Balanced 3 Phase Circuit

It is always better to solve the balanced three-phase circuits on the basis of each phase. When

the three-phase supply voltage is given without reference to the line or phase value, then it is

the line voltage which is taken into consideration.

The following steps are given below to solve the balanced three-phase circuits.

Step 1 – First of all draw the circuit diagram.

Step 2 – Determine XLP = XL/phase = 2πfL.

Step 3 – Determine XCP = XC/phase = 1/2πfC.

Step 4 – Determine XP = X/ phase = XL – XC

Step 5 – Determine ZP = Z/phase = √R2P + X2P

Step 6 – Determine cosϕ = RP/ZP; the power factor is lagging when XLP > XCP and it is

leading when XCP > XLP.

Step 7 – Determine the V phase.

For star connection VP = VL/√3 and for delta connection VP = VL

Step 8 – Determine IP = VP/ZP.

Step 9 – Now, determine the line current IL.

For star connection IL = IP and for delta connection IL = √3 IP

Step 10 – Determine the Active, Reactive and Apparent power.

Phase Sequence

In a three-phase system, the order in which the voltages attain their maximum positive value is

called Phase Sequence. There are three voltages or EMFs in the three-phase system with the

same magnitude, but the frequency is displaced by an angle of 120 deg electrically.

Taking an example, if the phases of any coil are named as R, Y, B then the Positive phase

sequence will be RYB, YBR, BRY also called clockwise sequence and similarly

the Negative phase sequence will be RBY, BYR, YRB respectively and known as an anti-

clockwise sequence.

37

It is essential because of the following reasons:

The parallel operation of the three-phase transformer or alternator is only possible when its

phase sequence is known.

The rotational direction of the three-phase induction motor depends upon its sequence of phase

on three-phase supply. And thus to reverse its direction the phase sequence of the supply given

to the motor has to be changed.

Delta Connection In a 3 Phase System

In Delta (Δ) or Mesh connection, the finished terminal of one winding is connected to start

terminal of the other phase and so on which gives a closed circuit. The three-line conductors

are run from the three junctions of the mesh called Line Conductors.

The connection in delta form is shown in the figure below:

To obtain the delta connections, a2 is connected with b1, b2 is connected with c1 and c2 is

connected with a1 as shown in the above figure. The three conductors R, Y and B are running

from the three junctions known as Line Conductors.

38

The current flowing through each phase is called Phase Current (Iph), and the current flowing

through each line conductor is called Line Current (IL).

The voltage across each phase is called Phase Voltage (Eph), and the voltage across two line

conductors is called Line Voltage (EL).

Relation Between Phase Voltage and Line Voltage in Delta Connection

To understand the relationship between the phase voltage and line voltage in the delta

connection, consider figure A shown below:

It is clear from the figure that the voltage across terminals 1 and 2 is the same as across the

terminals R and Y. Therefore,

Similarly,

: the phase voltages are

The line voltages are:

39

Hence, in delta connection line voltage is equal to phase voltage.

3.3 Relation Between Phase Current and Line Current in Delta Connection

As in the balanced system the three-phase current I12, I23 and I31 are equal in magnitude but

are displaced from one another by 120° electrical.

The phasor diagram is shown below:

Hence,

If we look at figure A, it is seen that the current is divided at every junction 1, 2 and 3.

Applying Kirchhoff’s Law at junction 1,

The Incoming currents are equal to outgoing currents.

And their vector difference will be given as:

40

The vector I12 is reversed and is added in the vector I31 to get the vector sum of I31 and –

I12 as shown above in the phasor diagram. Therefore,

As we know, IR = IL, therefore,

Similarly,

Hence, in delta connection line current is root three times of phase current.

This is all about Delta Connection In a 3 Phase System

3.4 Star Connection in a 3 Phase System

In the Star Connection, the similar ends (either start or finish) of the three windings are

connected to a common point called star or neutral point. The three-line conductors run from

the remaining three free terminals called line conductors.

The wires are carried to the external circuit, giving three-phase, three-wire star connected

systems. However, sometimes a fourth wire is carried from the star point to the external circuit,

called neutral wire, forming three-phase, four-wire star connected systems.

The star connection is shown in the diagram below:

41

Considering the above figure, the finish terminals a2, b2, and c2 of the three windings are

connected to form a star or neutral point. The three conductors named as R, Y and B run from

the remaining three free terminals as shown in the above figure.

The current flowing through each phase is called Phase current Iph, and the current flowing

through each line conductor is called Line Current IL. Similarly, the voltage across each

phase is called Phase Voltage Eph, and the voltage across two line conductors is known as

the Line Voltage EL.

3.5 Relation Between Phase Voltage and Line Voltage in Star Connection

The Star connection is shown in the figure below:

As the system is balanced, a balanced system means that in all the three phases, i.e., R, Y and

B, the equal amount of current flows through them. Therefore, the three voltages ENR,

ENY and ENB are equal in magnitude but displaced from one another by 120° electrical.

The Phasor Diagram of Star Connection is shown below:

42

The arrowheads on the EMFs and current indicate direction and not their actual direction at

any instant.

Now,

There are two-phase voltages between any two lines.

Tracing the loop NRYN

To find the vector sum of ENY and –ENR, we have to reverse the vector ENR and add it with

ENY as shown in the phasor diagram above.

Therefore,

43

3.6 Relation Between Phase Current and Line Current in Star Connection

The same current flows through phase winding as well as in the line conductor as it is

connected in series with the phase winding.

3.7 Advantages of 3 Phase Over Single Phase System

The three-phase system has three live conductors which supply the 440V to the large

consumers. While the single-phase system has one live conductor which is used for domestic

purposes. The following are the main advantages of 3 Phase system over Single Phase system.

44

Higher Rating

The rating, i.e. the output of a three-phase machine is nearly 1.5 times the rating (output) of a

single-phase machine of the same size.

Constant Power

In single-phase circuits, the power delivered is pulsating. Even when the voltage and current

are in phase, the power is zero twice in each cycle. Whereas, in the polyphase system, the

power delivered is almost constant when the loads are in a balanced condition.

Power Transmission Economics

The three-phase system requires only 75% of the weight of conducting material of that

required by the single-phase system to transmit the same amount of power over a fixed

distance at a given voltage.

Superiority of 3 Phase Induction Motors

The three phase induction motors have a wide field of applications in the industries because

of the following advantages are given below.

1. Three-phase induction motors are self-starting whereas the single-phase induction motor is

not self-starting. This means the 1 –phase motor has no starting torque and hence it needs

some auxiliary means to start at the initial stage.

2. The three Phase Induction motors have higher power factor and efficiency than that of a

single-phase induction motor.

Size and Weight of alternator

The 3 Phase Alternator is small in size and light in weight as compared to a single-phase

alternator.

Requirement of Copper and Aluminium

3 Phase system requires less copper and aluminium for the transmission system in

comparison to a single-phase transmission system.

Frequency of Vibration

In 3 phase motor, the frequency of vibrations is less as compared to single-phase motor

because in single-phase the power transferred is a function of current and varies constantly.

Dependency

A single-phase load can be efficiently fed by a 3 phase load or system, but 3 phase system

cannot depend or feed by a single-phase system.

Torque

A uniform or constant torque is produced in a 3 phase system, whereas in a single-phase

system pulsating torque is produced.

45

Unit- IV Network Theorems

4.1 Superposition Theorem

Superposition theorem states that in any linear, active, bilateral network having more than one

source, the response across any element is the sum of the responses obtained from each source

considered separately and all other sources are replaced by their internal resistance. The

superposition theorem is used to solve the network where two or more sources are present and

connected.

In other words, it can be stated as if a number of voltage or current sources are acting in a linear

network, the resulting current in any branch is the algebraic sum of all the currents that would

be produced in it when each source acts alone while all the other independent sources are

replaced by their internal resistances.

It is only applicable to the circuit which is valid for the ohm’s law (i.e., for the linear circuit).

Explanation of Superposition Theorem

Let us understand the superposition theorem with the help of an example. The circuit diagram

is shown below consists of two voltage sources V1 and V2.

First, take the source V1 alone and short circuit the V2 source as shown in the circuit diagram

below:

46

Here, the value of current flowing in each branch, i.e. i1’, i2’ and i3’ is calculated by the

following equations.

The difference between the above two equations gives the value of the current i3’

Now, activating the voltage source V2 and deactivating the voltage source V1 by short-

circuiting it, find the various currents, i.e. i1’’, i2’’, i3’’ flowing in the circuit diagram shown

below:

47

Here,

And the value of the current i3’’ will be calculated by the equation shown below:

As per the superposition theorem, the value of current i1, i2, i3 is now calculated as:

The direction of the current should be taken care of while finding the current in the various

branches.

4.2 Steps for Solving network by Superposition Theorem

Considering the circuit diagram A, let us see the various steps to solve the superposition

theorem:

48

Step 1 – Take only one independent source of voltage or current and deactivate the other

sources.

Step 2 – In the circuit diagram B shown above, consider the source E1 and replace the other

source E2 by its internal resistance. If its internal resistance is not given, then it is taken as zero

and the source is short-circuited.

Step 3 – If there is a voltage source than short circuit it and if there is a current source then just

open circuit it.

Step 4 – Thus, by activating one source and deactivating the other source find the current in

each branch of the network. Taking the above example find the current I1’, I2’and I3’.

Step 5 – Now consider the other source E2 and replace the source E1 by its internal resistance

r1 as shown in the circuit diagram C.

Step 6 – Determine the current in various sections, I1’’, I2’’ and I3’’.

Step 7 – Now to determine the net branch current utilizing the superposition theorem, add the

currents obtained from each individual source for each branch.

Step 8 – If the current obtained by each branch is in the same direction then add them and if it

is in the opposite direction, subtract them to obtain the net current in each branch.

The actual flow of current in the circuit C will be given by the equations shown below

49

Thus, in this way, we can solve superposition theorem.

4.3 Reciprocity Theorem

Reciprocity Theorem states that – In any branch of a network or circuit, the current due to a

single source of voltage (V) in the network is equal to the current through that branch in which

the source was originally placed when the source is again put in the branch in which the current

was originally obtained. This theorem is used in the bilateral linear network which consists of

bilateral components.

In simple words, we can state the reciprocity theorem as when the places of voltage and current

source in any network are interchanged the amount or magnitude of current and voltage flowing

in the circuit remains the same.

This theorem is used for solving many DC and AC network which have many applications in

electromagnetism electronics. These circuits do not have any time-varying element.

Explanation of Reciprocity Theorem

The location of the voltage source and the current source may be interchanged without a

change in current. However, the polarity of the voltage source should be identical with the

direction of the branch current in each position.

The Reciprocity Theorem is explained with the help of the circuit diagram shown below

The various resistances R1, R2, R3 is connected in the circuit diagram above with a voltage

source (V) and a current source (I). It is clear from the figure above that the voltage source and

current sources are interchanged for solving the network with the help of Reciprocity Theorem.

The limitation of this theorem is that it is applicable only to single-source networks and not in

the multi-source network. The network where reciprocity theorem is applied should be linear

50

and consist of resistors, inductors, capacitors and coupled circuits. The circuit should not

have any time-varying elements.

Steps for Solving a Network Utilizing Reciprocity Theorem

Step 1 – Firstly, select the branches between which reciprocity has to be established.

Step 2 – The current in the branch is obtained using any conventional network analysis

method.

Step 3 – The voltage source is interchanged between the branch which is selected.

Step 4 – The current in the branch where the voltage source was existing earlier is calculated.

Step 5 – Now, it is seen that the current obtained in the previous connection, i.e., in step 2 and

the current which is calculated when the source is interchanged, i.e., in step 4 are identical to

each other.

4.4 Thevenin’s Theorem

Thevenin’s Theorem states that any complicated network across its load terminals can be

substituted by a voltage source with one resistance in series. This theorem helps in the study of

the variation of current in a particular branch when the resistance of the branch is varied while

the remaining network remains the same.

For example in designing electrical and electronics circuits.

A more general statement of Thevenin’s Theorem is that any linear active network consisting

of independent or dependent voltage and current source and the network elements can be

replaced by an equivalent circuit having a voltage source in series with a resistance.

Where the voltage source being the open-circuited voltage across the open-circuited load

terminals and the resistance being the internal resistance of the source.

In other words, the current flowing through a resistor connected across any two terminals of a

network by an equivalent circuit having a voltage source Eth in series with a resistor Rth.

Where Eth is the open-circuit voltage between the required two terminals called the Thevenin

voltage and the Rth is the equivalent resistance of the network as seen from the two-terminal

with all other sources replaced by their internal resistances called Thevenin resistance.

Explanation of Thevenin’s Theorem

The Thevenin’s statement is explained with the help of a circuit shown below:

51

Let us consider a simple DC circuit as shown in the figure above, where we have to find the

load current IL by the Thevenin’s theorem.

In order to find the equivalent voltage source, rL is removed from the circuit as shown in the

figure below and Voc or VTH is calculated.

Now, to find the internal resistance of the network (Thevenin’s resistance or equivalent

resistance) in series with the open-circuit voltage VOC , also known as Thevenin’s

voltage VTH, the voltage source is removed or we can say it is deactivated by a short circuit

(as the source does not have any internal resistance) as shown in the figure below:

52

,

As per Thevenin’s Statement, the load current is determined by the circuit shown above and

the equivalent Thevenin’s circuit is obtained.

The load current IL is given as:

Where,

53

VTH is the Thevenin’s equivalent voltage. It is an open circuit voltage across the terminal

AB known as load terminal

RTH is the Thevenin’s equivalent resistance, as seen from the load terminals where all the

sources are replaced by their internal impedance

rL is the load resistance

Steps for Solving Thevenin’s Theorem

Step 1 – First of all remove the load resistance rL of the given circuit.

Step 2 – Replace all the sources by their internal resistance.

Step 3 – If sources are ideal then short circuit the voltage source and open circuit the current

source.

Step 4 – Now find the equivalent resistance at the load terminals, known as Thevenin’s

Resistance (RTH).

Step 5 – Draw the Thevenin’s equivalent circuit by connecting the load resistance and after

that determine the desired response.

This theorem is possibly the most extensively used networks theorem. It is applicable where

it is desired to determine the current through or voltage across any one element in a network.

Thevenin’s Theorem is an easy way to solve a complicated network.

4.5 Norton’s Theorem

Norton’s Theorem states that – A linear active network consisting of the independent or

dependent voltage source and current sources and the various circuit elements can be

substituted by an equivalent circuit consisting of a current source in parallel with a resistance.

The current source being the short-circuited current across the load terminal and the resistance

being the internal resistance of the source network.

The Norton’s theorems reduce the networks equivalent to the circuit having one current source,

parallel resistance and load. Norton’s theorem is the converse of Thevenin’s Theorem. It

consists of the equivalent current source instead of an equivalent voltage source as in

Thevenin’s theorem.

The determination of internal resistance of the source network is identical in both the theorems.

In the final stage that is in the equivalent circuit, the current is placed in parallel to the internal

resistance in Norton’s Theorem whereas in Thevenin’s Theorem the equivalent voltage source

is placed in series with the internal resistance.

Explanation of Norton’s Theorem

To understand Norton’s Theorem in detail, let us consider a circuit diagram given below

54

In order to find the current through the load resistance IL as shown in the circuit diagram

above, the load resistance has to be short-circuited as shown in the diagram below:

Now, the value of current I flowing in the circuit is found out by the equation

And the short-circuit current ISC is given by the equation shown below:

55

Now the short circuit is removed, and the independent source is deactivated as shown in the

circuit diagram below and the value of the internal resistance is calculated by:

As per Norton’s Theorem, the equivalent source circuit would contain a current source in

parallel to the internal resistance, the current source being the short-circuited current across

the shorted terminals of the load resistor. The Norton’s Equivalent circuit is represented as

Finally, the load current IL calculated by the equation shown below

Where,

IL is the load current

56

Isc is the short circuit current

Rint is the internal resistance of the circuit

RL is the load resistance of the circuit

Steps for Solving a Network Utilizing Norton’s Theorem

Step 1 – Remove the load resistance of the circuit.

Step 2 – Find the internal resistance Rint of the source network by deactivating the constant

sources.

Step 3 – Now short the load terminals and find the short circuit current ISC flowing through

the shorted load terminals using conventional network analysis methods.

Step 4 – Norton’s equivalent circuit is drawn by keeping the internal resistance Rint in

parallel with the short circuit current ISC.

Step 5 – Reconnect the load resistance RL of the circuit across the load terminals and find the

current through it known as load current IL.

4.6 Maximum Power Transfer Theorem

Maximum Power Transfer Theorem states that – A resistive load, being connected to a DC

network, receives maximum power when the load resistance is equal to the internal resistance

known as (Thevenin’s equivalent resistance) of the source network as seen from the load

terminals. The Maximum Power Transfer theorem is used to find the load resistance for which

there would be the maximum amount of power transfer from the source to the load.

The maximum power transfer theorem is applied to both the DC and AC circuit. The only

difference is that in the AC circuit the resistance is substituted by the impedance.

The maximum power transfer theorem finds their applications in communication systems

which receive low strength signal. It is also used in speaker for transferring the maximum

power from an amplifier to the speaker.

Explanation of Maximum Power Transfer Theorem

A variable resistance RL is connected to a DC source network as shown in the circuit diagram

in figure A below and the figure B represents the Thevenin’s voltage VTH and Thevenin’s

resistance RTH of the source network.

The aim of the Maximum Power Transfer theorem is to determine the value of load resistance

RL, such that it receives maximum power from the DC source

57

Considering figure B the value of current will be calculated by the equation shown below

While the power delivered to the resistive load is given by the equation

Putting the value of I from the equation (1) in the equation (2) we will get

PL can be maximized by varying RL and hence, maximum power can be delivered when

(dPL/dRL) = 0

However,

58

But as we know, (dPL/dRL) = 0

Therefore,

Which gives

Hence, it is proved that power transfer from a DC source network to a resistive network is

maximum when the internal resistance of the DC source network is equal to the load resistance.

Again, with RTH = RL, the system is perfectly matched to the load and the source, thus, the

power transfer becomes maximum, and this amount of power Pmax can be obtained by the

equation shown below:

Equation (3) gives the power which is consumed by the load. The power transfer by the source

will also be the same as the power consumed by the load, i.e. equation (3), as the load power

and the source power being the same.

Thus, the total power supplied is given by the equation

During Maximum Power Transfer the efficiency ƞ becomes:

59

The concept of Maximum Power Transfer theorem is that by making the source resistance

equal to the load resistance, which has wide application in communication circuits where the

magnitude of power transfer is sufficiently small. To achieve maximum power transfer, the

source and the load resistance are matched and with this, efficiency becomes 50% with the

flow of maximum power from the source to the load.

In the Electrical Power Transmission system, the load resistance being sufficiently greater than

the source resistance, it is difficult to achieve the condition of maximum power transfer.

In power system emphasis is given to keep the voltage drops and the line losses to a minimum

value and hence the operation of the power system, operating with bulk power transmission

capability, becomes uneconomical if it is operating with only 50% efficiency just for achieving

maximum power transfer.

Hence, in the electrical power transmission system, the criterion of maximum power transfer

is very rarely used.

Steps for Solving Network Using Maximum Power Transfer Theorem

Following steps are used to solve the problem by Maximum Power Transfer theorem

Step 1 – Remove the load resistance of the circuit.

Step 2 – Find the Thevenin’s resistance (RTH) of the source network looking through the

open-circuited load terminals.

Step 3 – As per the maximum power transfer theorem, this RTH is the load resistance of the

network, i.e., RL = RTH that allows maximum power transfer.

Step 4 – Maximum Power Transfer is calculated by the equation shown below

4.7 Millman’s Theorem

The Millman’s Theorem states that – when a number of voltage sources (V1, V2, V3………

Vn) are in parallel having internal resistance (R1, R2, R3………….Rn) respectively, the

arrangement can replace by a single equivalent voltage source V in series with an equivalent

series resistance R. In other words; it determines the voltage across the parallel branches of

the circuit, which have more than one voltage sources, i.e., reduces the complexity of the

electrical circuit.

This Theorem is given by Jacob Millman. The utility of Millman’s Theorem is that the number

of parallel voltage sources can be reduced to one equivalent source. It is applicable only to

60

solve the parallel branch with one resistance connected to one voltage source or current source.

It is also used in solving network having an unbalanced bridge circuit.

As per Millman’s Theorem

Explanation of Millman’s Theorem

Assuming a DC network of numerous parallel voltage sources with internal resistances

supplying power to a load resistance RL as shown in the figure below

61

Let I represent the resultant current of the parallel current sources while G the equivalent

conductance as shown in the figure below

Next, the resulting current source is converted to an equivalent voltage source as shown in the

figure below

62

Positive (+) and negative (-) sign appeared to include the cases where the sources may not be

supplying current in the same direction.

And as we know,

I = V/R, and we can also write R = I/G as G = I/R

So the equation can be written as

Where R is the equivalent resistance connected to the equivalent voltage source in series.

Thus, the final equation becomes

Steps for Solving Millman’s Theorem

Following steps are used to solve the network by Millman’s Theorem

Step 1 – Obtain the conductance (G1, G2,….) of each voltage source (V1, V2,….).

Step 2 – Find the value of equivalent conductance G by removing the load from the network.

Step 3 – Now, apply Millman’s Theorem to find the equivalent voltage source V by the

equation shown below

63

Step 4 – Determine the equivalent series resistance (R) with the equivalent voltage sources

(V) by the equation

Step 5 – Find the current IL flowing in the circuit across the load resistance RL by the

equation

4.8 Tellegen’s Theorem

Tellegen’s Theorem states that the summation of power delivered is zero for each branch of

any electrical network at any instant of time. It is mainly applicable for designing the filters in

signal processing.

It is also used in complex operating systems for regulating stability. It is mostly used in the

chemical and biological system and for finding the dynamic behaviour of the physical network.

Tellegen’s theorem is independent of the network elements. Thus, it is applicable for any

lump system that has linear, active, passive and time-variant elements. Also, the theorem is

convenient for the network which follows Kirchoff’s current law and Kirchoff’s voltage law.

Explanation of Tellegen’s Theorem

Tellegen’s Theorem can also be stated in another word as, in any linear, nonlinear, passive,

active, time-variant or time-invariant network the summation of power (instantaneous or

complex power of sources) is zero.

Thus, for the Kth branch, this theorem states that:

Where,

n is the number of branches

vK is the voltage in the branch

iK is the current flowing through the branch.

64

Let,

Equation (1) shows the Kth branch through current

vK is the voltage drop in branch K and is given as:

Where vp and vq are the respective node voltage at p and q nodes.

We have,

Also,

Obviously

Summing the above two equations (2) and (3), we get

Such equations can be written for every branch of the network.

65

Assuming n branches, the equation will be:

However, according to the Kirchhoff’s current law (KCL), the algebraic sum of currents at

each node is equal to zero.

Therefore,

Thus, from the above equation (4) finally, we obtain

Thus, it has been observed that the sum of power delivered to a closed network is zero. This

proves the Tellegen’s theorem and also proves the conservation of power in any electrical

network.

It is also evident that the sum of power delivered to the network by an independent source is

equal to the sum of power absorbed by all passive elements of the network.

Steps for Solving Networks Using Tellegen’s Theorem

The following steps are given below to solve any electrical network by Tellegen’s theorem:

Step 1 – In order to justify this theorem in an electrical network, the first step is to find the

branch voltage drops.

Step 2 – Find the corresponding branch currents using conventional analysis methods.

Step 3 – Tellegen’s theorem can then be justified by summing the products of all branch

voltages and currents.

For example, if a network having some branches “b” then:

66

Now, if the set of voltages and currents is taken, corresponding the two different instants of

time, t1 and t2, then Tellegen’s theorem is also applicable where we get the equation as

shown below:

Application of Tellegen’s Theorem

The various applications of the Tellegen’s theorem are as follows:

It is used in the digital signal processing system for designing filters.

In the area of the biological and chemical process.

In topology and structure of reaction network analysis.

The theorem is used in chemical plants and oil industries to determine the stability of any

complex systems.

4.9 Compensation Theorem

Compensation Theorem states that in a linear time-invariant network when the resistance (R)

of an uncoupled branch, carrying a current (I), is changed by (ΔR), then the currents in all the

branches would change and can be obtained by assuming that an ideal voltage source of (VC)

has been connected such that VC = I (ΔR) in series with (R + ΔR) when all other sources in

the network are replaced by their internal resistances.

In Compensation Theorem, the source voltage (VC) opposes the original current. In simple

words, compensation theorem can be stated as – the resistance of any network can be

replaced by a voltage source, having the same voltage as the voltage drop across the

resistance which is replaced.

67

Explanation

Let us assume a load RL be connected to a DC source network whose Thevenin’s equivalent

gives V0 as the Thevenin’s voltage and RTH as the Thevenin’s resistance as shown in the

figure below:

Here,

Let the load resistance RL be changed to (RL + ΔRL). Since the rest of the circuit remains

unchanged, the Thevenin’s equivalent network remains the same as shown in the circuit

diagram below:

Here,

68

The change of current being termed as ΔI

Therefore,

Putting the value of I’ and I from the equation (1) and (2) in the equation (3) we will get the

following equation:

Now, putting the value of I from the equation (1) in equation (4), we will get the following

equation:

As we know, VC = I ΔRL and is known as compensating voltage.

Therefore, equation (5) becomes,

Hence, Compensation theorem tells that with the change of branch resistance, branch currents

changes and the change is equivalent to an ideal compensating voltage source in series with

the branch opposing the original current, where all other sources in the network being

replaced by their internal resistances.

69

Unit- V Network Topology

Network topology is a graphical representation of electric circuits. It is useful for analyzing

complex electric circuits by converting them into network graphs. Network topology is also

called as Graph theory.

Basic Terminology of Network Topology

Now, let us discuss about the basic terminology involved in this network topology.

5.1 Graph

Network graph is simply called as graph. It consists of a set of nodes connected by branches.

In graphs, a node is a common point of two or more branches. Sometimes, only a single branch

may connect to the node. A branch is a line segment that connects two nodes.

Any electric circuit or network can be converted into its equivalent graph by replacing the

passive elements and voltage sources with short circuits and the current sources with open

circuits. That means, the line segments in the graph represent the branches corresponding to

either passive elements or voltage sources of electric circuit.

Example

Let us consider the following electric circuit.

In the above circuit, there are four principal nodes and those are labelled with 1, 2, 3, and 4.

There are seven branches in the above circuit, among which one branch contains a 20 V voltage

source, another branch contains a 4 A current source and the remaining five branches contain

resistors having resistances of 30 Ω, 5 Ω, 10 Ω, 10 Ω and 20 Ω respectively.

An equivalent graph corresponding to the above electric circuit is shown in the following

figure.

70

In the above graph, there are four nodes and those are labelled with 1, 2, 3 & 4 respectively.

These are same as that of principal nodes in the electric circuit. There are six branches in the

above graph and those are labelled with a, b, c, d, e & f respectively.

In this case, we got one branch less in the graph because the 4 A current source is made as open

circuit, while converting the electric circuit into its equivalent graph.

From this Example, we can conclude the following points −

The number of nodes present in a graph will be equal to the number of principal nodes present

in an electric circuit.

The number of branches present in a graph will be less than or equal to the number of branches

present in an electric circuit.

5.2 Types of Graphs

Following are the types of graphs −

Connected Graph

Unconnected Graph

Directed Graph

Undirected Graph

Now, let us discuss these graphs one by one.

Connected Graph

If there exists at least one branch between any of the two nodes of a graph, then it is called as

a connected graph. That means, each node in the connected graph will be having one or more

branches that are connected to it. So, no node will present as isolated or separated.

The graph shown in the previous Example is a connected graph. Here, all the nodes are

connected by three branches.

Unconnected Graph

71

If there exists at least one node in the graph that remains unconnected by even single branch,

then it is called as an unconnected graph. So, there will be one or more isolated nodes in an

unconnected graph.

Consider the graph shown in the following figure.

In this graph, the nodes 2, 3, and 4 are connected by two branches each. But, not even a

single branch has been connected to the node 1. So, the node 1 becomes an isolated node.

Hence, the above graph is an unconnected graph.

5.3 Directed Graph

If all the branches of a graph are represented with arrows, then that graph is called as

a directed graph. These arrows indicate the direction of current flow in each branch. Hence,

this graph is also called as oriented graph.

Consider the graph shown in the following figure.

In the above graph, the direction of current flow is represented with an arrow in each branch.

Hence, it is a directed graph.

Undirected Graph

72

If the branches of a graph are not represented with arrows, then that graph is called as

an undirected graph. Since, there are no directions of current flow, this graph is also called as

an unoriented graph.

The graph that was shown in the first Example of this chapter is an unoriented graph, because

there are no arrows on the branches of that graph.

5.4 Subgraph and its Types

A part of the graph is called as a subgraph. We get subgraphs by removing some nodes and/or

branches of a given graph. So, the number of branches and/or nodes of a subgraph will be

less than that of the original graph. Hence, we can conclude that a subgraph is a subset of a

graph.

Following are the two types of subgraphs.

Tree

Co-Tree

Tree

Tree is a connected subgraph of a given graph, which contains all the nodes of a graph. But,

there should not be any loop in that subgraph. The branches of a tree are called as twigs.

Consider the following connected subgraph of the graph, which is shown in the Example of

the beginning of this chapter.

This connected subgraph contains all the four nodes of the given graph and there is no loop.

Hence, it is a Tree.

This Tree has only three branches out of six branches of given graph. Because, if we consider

even single branch of the remaining branches of the graph, then there will be a loop in the

above connected subgraph. Then, the resultant connected subgraph will not be a Tree.

From the above Tree, we can conclude that the number of branches that are present in a Tree

should be equal to n - 1 where ‘n’ is the number of nodes of the given graph.

Co-Tree

Co-Tree is a subgraph, which is formed with the branches that are removed while forming a

Tree. Hence, it is called as Complement of a Tree. For every Tree, there will be a

73

corresponding Co-Tree and its branches are called as links or chords. In general, the links are

represented with dotted lines.

The Co-Tree corresponding to the above Tree is shown in the following figure.

This Co-Tree has only three nodes instead of four nodes of the given graph, because Node 4 is

isolated from the above Co-Tree. Therefore, the Co-Tree need not be a connected subgraph.

This Co-Tree has three branches and they form a loop.

The number of branches that are present in a co-tree will be equal to the difference between the

number of branches of a given graph and the number of twigs. Mathematically, it can be written

as

l=b−(n−1)l=b−(n−1)

l=b−n+1l=b−n+1

Where,

l is the number of links.

b is the number of branches present in a given graph.

n is the number of nodes present in a given graph.

If we combine a Tree and its corresponding Co-Tree, then we will get the original graph as

shown below.

74

The Tree branches d, e & f are represented with solid lines. The Co-Tree branches a, b & c

are represented with dashed lines.

5.5 Voltage Sources

A Voltage Source is a device that generates an exact output voltage which, in theory, does not

change regardless of the load current

there are two types of elements within an electrical or electronics circuit: passive

elements and active elements. An active element is one that is capable of continuously

supplying energy to a circuit, such as a battery, a generator, an operational amplifier, etc. A

passive element on the other hand are physical elements such as resistors, capacitors, inductors,

etc, which cannot generate electrical energy by themselves but only consume it.

The types of active circuit elements that are most important to us are those that supply electrical

energy to the circuits or network connected to them. These are called “electrical sources” with

the two types of electrical sources being the voltage source and the current source. The current

source is usually less common in circuits than the voltage source, but both are used and can be

regarded as complements of each other.

An electrical supply or simply, “a source”, is a device that supplies electrical power to a circuit

in the form of a voltage source or a current source. Both types of electrical sources can be

classed as a direct (DC) or alternating (AC) source in which a constant voltage is called a DC

voltage and one that varies sinusoidally with time is called an AC voltage. So for example,

batteries are DC sources and the 230V wall socket or mains outlet in your home is an AC

source.

We said earlier that electrical sources supply energy, but one of the interesting characteristic

of an electrical source, is that they are also capable of converting non-electrical energy into

electrical energy and vice versa. For example, a battery converts chemical energy into electrical

energy, while an electrical machine such as a DC generator or an AC alternator converts

mechanical energy into electrical energy.

Renewable technologies can convert energy from the sun, the wind, and waves into electrical

or thermal energy. But as well as converting energy from one source to another, electrical

sources can both deliver or absorb energy allowing it to flow in both directions.

Another important characteristic of an electrical source and one which defines its operation,

are its I-V characteristics. The I-V characteristic of an electrical source can give us a very nice

pictorial description of the source, either as a voltage source and a current source as shown.

75

Electrical Sources

Electrical sources, both as a voltage source or a current source can be classed as being

either independent (ideal) or dependent, (controlled) that is whose value depends upon a

voltage or current elsewhere within the circuit, which itself can be either constant or time-

varying.

When dealing with circuit laws and analysis, electrical sources are often viewed as being

“ideal”, that is the source is ideal because it could theoretically deliver an infinite amount of

energy without loss thereby having characteristics represented by a straight line. However, in

real or practical sources there is always a resistance either connected in parallel for a current

source, or series for a voltage source associated with the source affecting its output.

The Voltage Source

A voltage source, such as a battery or generator, provides a potential difference (voltage)

between two points within an electrical circuit allowing current to flowing around it.

Remember that voltage can exist without current. A battery is the most common voltage source

for a circuit with the voltage that appears across the positive and negative terminals of the

source being called the terminal voltage.

Ideal Voltage Source

76

An ideal voltage source is defined as a two terminal active element that is capable of supplying

and maintaining the same voltage, (v) across its terminals regardless of the current, (i) flowing

through it. In other words, an ideal voltage source will supply a constant voltage at all times

regardless of the value of the current being supplied producing an I-V characteristic represented

by a straight line.

Then an ideal voltage source is known as an Independent Voltage Source as its voltage does

not depend on either the value of the current flowing through the source or its direction but is

determined solely by the value of the source alone. So for example, an automobile battery has

a 12V terminal voltage that remains constant as long as the current through it does not become

to high, delivering power to the car in one direction and absorbing power in the other direction

as it charges.

On the other hand, a Dependent Voltage Source or controlled voltage source, provides a

voltage supply whose magnitude depends on either the voltage across or current flowing

through some other circuit element. A dependent voltage source is indicated with a diamond

shape and are used as equivalent electrical sources for many electronic devices, such as

transistors and operational amplifiers.

Connecting Voltage Sources Together

Ideal voltage sources can be connected together in both parallel or series the same as for any

circuit element. Series voltages add together while parallel voltages have the same value.

Note that unequal ideal voltage sources cannot be connected directly together in parallel.

Voltage Source in Parallel

While not best practice for circuit analysis, ideal voltage sources can be connected in parallel

provided they are of the same voltage value. Here in this example, two 10 volt voltage source

are combined to produce 10 volts between terminals A and B. Ideally, there would be just one

single voltage source of 10 volts given between terminals A and B.

77

What is not allowed or is not best practice, is connecting together ideal voltage sources that

have different voltage values as shown, or are short-circuited by an external closed loop or

branch.

Badly Connected Voltage Sources

However, when dealing with circuit analysis, voltage sources of different values can be used

providing there are other circuit elements in between them to comply with Kirchoff’s Voltage

Law, KVL.

Unlike parallel connected voltage sources, ideal voltage sources of different values can be

connected together in series to form a single voltage source whose output will be the algebraic

addition or subtraction of the voltages used. Their connection can be as: series-aiding or series-

opposing voltages as shown.

5.6 Voltage Source in Series

Series aiding voltage sources are series connected sources with their polarities connected so

that the plus terminal of one is connected to the negative terminal of the next allowing current

to flow in the same direction. In the example above, the two voltages of 10V and 5V of the first

circuit can be added, for a VS of 10 + 5 = 15V. So the voltage across terminals A and B is 15

volts.

Series opposing voltage sources are series connected sources which have their polarities

connected so that the plus terminal or the negative terminals are connected together as shown

in the second circuit above. The net result is that the voltages are subtracted from each other.

Then the two voltages of 10V and 5V of the second circuit are subtracted with the smaller

voltage subtracted from the larger voltage. Resulting in a VS of 10 – 5 = 5V.

78

The polarity across terminals A and B is determined by the larger polarity of the voltage

sources, in this example terminal A is positive and terminal B is negative resulting in +5 volts.

If the series-opposing voltages are equal, the net voltage across A and B will be zero as one

voltage balances out the other. Also any currents (I) will also be zero, as without any voltage

source, current can not flow.

Voltage Source Example No1

Two series aiding ideal voltage sources of 6 volts and 9 volts respectively are connected

together to supply a load resistance of 100 Ohms. Calculate: the source voltage, VS, the load

current through the resistor, IR and the total power, P dissipated by the resistor. Draw the

circuit.

Thus, VS = 15V, IR = 150mA or 0.15A, and PR = 2.25W.

5.7 Practical Voltage Source

We have seen that an ideal voltage source can provide a voltage supply that is independent of

the current flowing through it, that is, it maintains the same voltage value always. This idea

may work well for circuit analysis techniques, but in the real world voltage sources behave a

little differently as for a practical voltage source, its terminal voltage will actually decrease

with an increase in load current.

As the terminal voltage of an ideal voltage source does not vary with increases in the load

current, this implies that an ideal voltage source has zero internal resistance, RS = 0. In other

words, it is a resistor less voltage source. In reality all voltage sources have a very small internal

resistance which reduces their terminal voltage as they supply higher load currents.

For non-ideal or practical voltage sources such as batteries, their internal resistance (RS)

produces the same effect as a resistance connected in series with an ideal voltage source as

these two series connected elements carry the same current as shown.

Ideal and Practical Voltage Source

79

You may have noticed that a practical voltage source closely resembles that of a Thevenin’s

equivalent circuit as Thevenin’s theorem states that “any linear network containing resistances

and sources of emf and current may be replaced by a single voltage source, VS in series with a

single resistance, RS“. Note that if the series source resistance is low, the voltage source is

ideal. When the source resistance is infinite, the voltage source is open-circuited.

In the case of all real or practical voltage sources, this internal resistance, RS no matter how

small has an effect on the I-V characteristic of the source as the terminal voltage falls off with

an increase in load current. This is because the same load current flows through RS.

Ohms law tells us that when a current, (i) flows through a resistance, a voltage drop is produce

across the same resistance. The value of this voltage drop is given as i*RS. Then VOUT will

equal the ideal voltage source, VS minus the i*RS voltage drop across the resistor. Remember

that in the case of an ideal source voltage, RS is equal to zero as there is no internal resistance,

therefore the terminal voltage is same as VS.

Then the voltage sum around the loop given by Kirchoff’s voltage law, KVL is:

VOUT = VS – i*RS. This equation can be plotted to give the I-V characteristics of the actual

output voltage. It will give a straight line with a slope –RS which intersects the vertical voltage

axis at the same point as VS when the current i = 0 as shown.

5.8 Practical Voltage Source Characteristics

Therefore, all ideal voltage sources will have a straight line I-V characteristic but non-ideal or

real practical voltage sources will not but instead will have an I-V characteristic that is slightly

angled down by an amount equal to i*RS where RS is the internal source resistance (or

impedance). The I-V characteristics of a real battery provides a very close approximation of an

ideal voltage source since the source resistance RS is usually quite small.

The decrease in the angle of the slope of the I-V characteristics as the current increases is

known as regulation. Voltage regulation is an important measure of the quality of a practical

voltage source as it measures the variation in terminal voltage between no load, that is when

IL = 0, (an open-circuit) and full load, that is when IL is at maximum, (a short-circuit).

Voltage Source Example No2

A battery supply consists of an ideal voltage source in series with an internal resistor. The

voltage and current measured at the terminals of the battery were found to be VOUT1 = 130V

at 10A, and VOUT2 = 100V at 25A. Calculate the voltage rating of the ideal voltage source

and the value of its internal resistance. Draw the I-V characteristics.

Firstly lets define in simple “simultaneous equation form“, the two voltage and current outputs

of the battery supply given as: VOUT1 and VOUT2.

80

As with have the voltages and currents in a simultaneous equation form, to find VS we will

first multiply VOUT1 by five, (5) and VOUT2 by two, (2) as shown to make the value of the

two currents, (i) the same for both equations.

Having made the co-efficients for RS the same by multiplying through with the previous

constants, we now multiply the second equation VOUT2 by minus one, (-1) to allow for the

subtraction of the two equations so that we can solve for VS as shown.

81

Knowing that the ideal voltage source, VS is equal to 150 volts, we can use this value for

equation VOUT1 (or VOUT2 if so wished) and solve to find the series resistance, RS.

Then for our simple example, the batteries internal voltage source is calculated as: VS = 150

volts, and its internal resistance as: RS = 2Ω. The I-V characteristics of the battery are given

as:

Battery I-V Characteristics

5.9 Dependent Voltage Source

Unlike an ideal voltage source which produces a constant voltage across its terminals regardless

of what is connected to it, a controlled or dependent voltage source changes its terminal voltage

depending upon the voltage across, or the current through, some other element connected to

the circuit, and as such it is sometimes difficult to specify the value of a dependent voltage

source, unless you know the actual value of the voltage or current on which it depends.

Dependent voltage sources behave similar to the electrical sources we have looked at so far,

both practical and ideal (independent) the difference this time is that a dependent voltage source

82

can be controlled by an input current or voltage. A voltage source that depends on a voltage

input is generally referred to as a Voltage Controlled Voltage Source or VCVS. A voltage

source that depends on a current input is referred too as a Current Controlled Voltage

Source or CCVS.

Ideal dependent sources are commonly used in the analysing the input/output characteristics or

the gain of circuit elements such as operational amplifiers, transistors and integrated circuits.

Generally, an ideal voltage dependent source, either voltage or current controlled is designated

by a diamond-shaped symbol as shown.

Dependent Voltage Source Symbols

An ideal dependent voltage-controlled voltage source, VCVS, maintains an output voltage

equal to some multiplying constant (basically an amplification factor) times the controlling

voltage present elsewhere in the circuit. As the multiplying constant is, well, a constant, the

controlling voltage, VIN will determine the magnitude of the output voltage, VOUT. In other

words, the output voltage “depends” on the value of input voltage making it a dependent

voltage source and in many ways, an ideal transformer can be thought of as a VCVS device

with the amplification factor being its turns ratio.

Then the VCVS output voltage is determined by the following equation: VOUT = μVIN. Note

that the multiplying constant μ is dimensionless as it is purely a scaling factor because

μ = VOUT/VIN, so its units will be volts/volts.

An ideal dependent current-controlled voltage source, CCVS, maintains an output voltage

equal to some multiplying constant (rho) times a controlling current input generated elsewhere

within the connected circuit. Then the output voltage “depends” on the value of the input

current, again making it a dependent voltage source.

As a controlling current, IIN determines the magnitude of the output voltage, VOUT times the

magnification constant ρ (rho), this allows us to model a current-controlled voltage source as a

trans-resistance amplifier as the multiplying constant, ρ gives us the following equation:

VOUT = ρIIN. This multiplying constant ρ (rho) has the units of Ohm’s because

ρ = VOUT/IIN, and its units will therefore be volts/amperes.

Voltage Source Summary

We have seen here that a Voltage Source can be either an ideal independent voltage source, or

a controlled dependent voltage source. Independent voltage sources supply a constant voltage

83

that does not depend on any other quantity within the circuit. Ideal independent sources can be

batteries, DC generators or time-varying AC voltage supplies from alternators.

Independent voltage sources can be modelled as either an ideal voltage source, (RS = 0) where

the output is constant for all load currents, or a non-ideal or practical, such as a battery with a

resistance connected in series with the circuit to represent the internal resistance of the source.

Ideal voltage sources can be connected together in parallel only if they are of the same voltage

value. Series-aiding or series-opposing connections will affect the output value.

Also for solving circuit analysis and complex theorems, voltage sources become short-circuited

sources making their voltage equal to zero to help solve the network. Note also that voltage

sources are capable of both delivering or absorbing power.

Ideal dependent voltage sources represented by a diamond-shaped symbol, are dependent on,

and are proportional too an external controlling voltage or current. The multiplying constant,

μ for a VCVS has no units, while the multiplying constant ρ for a CCVS has units of Ohm’s.

A dependent voltage source is of great interest to model electronic devices or active devices

such as operational amplifiers and transistors that have gain.

In the next tutorial about electrical sources, we will look at the compliment of the voltage

source, that is the current source and see that current sources can also be classed as dependent

or independent electrical sources.

5.10 Principle of Duality:

Principle of duality in context of electrical networks states that

A dual of a relationship is one in which current and voltage are interchangeable

Two networks are dual to each other if one has mesh equation numerically identical to others

node equation

List of Dual Pairs:

For evaluating a dual network, you should follow these points

The number of meshes in a network is equal to number of nodes in its dual network

The impedance of a branch common to two meshes must be equal to admittance between two

nodes in the dual network

Voltage source common to both loops must be replaced by a current source between two nodes

Open switch in a network is replaced by a closed switch in its dual network or vice versa

84

Formation of Dual Networks:

The principle of duality is applicable to planar circuits only. Carefully read the points stated

below, follow each step and draw the dual circuit

Place a dot within each loop, these dots will become nodes of the dual network

Place a dot outside of the network, this dot will be the ground/datum node of the dual network

Carefully draw lines between nodes such that each line cuts only one element

If an element exclusively present in a loop, then connect the dual element in between node and

ground/datum node

If an element is common in between two loops, then dual element is placed in between two

nodes

Branch containing active source, consider as a separate branch

Now to determine polarity of voltage source and direction of current sources, consider voltage

source producing clockwise current in a loop. Its dual current source will have a current

direction from ground to non-reference node

Nodal Voltage Analysis

Nodal Voltage Analysis finds the unknown voltage drops around a circuit between different

nodes that provide a common connection for two or more circuit components

85

Nodal Voltage Analysis complements the previous mesh analysis in that it is equally powerful

and based on the same concepts of matrix analysis. As its name implies, Nodal Voltage

Analysis uses the “Nodal” equations of Kirchhoff’s first law to find the voltage potentials

around the circuit.

So by adding together all these nodal voltages the net result will be equal to zero. Then, if there

are “n” nodes in the circuit there will be “n-1” independent nodal equations and these alone are

sufficient to describe and hence solve the circuit.

At each node point write down Kirchhoff’s first law equation, that is: “the currents entering a

node are exactly equal in value to the currents leaving the node” then express each current in

terms of the voltage across the branch. For “n” nodes, one node will be used as the reference

node and all the other voltages will be referenced or measured with respect to this common

node.

For example, consider the circuit from the previous section.

Nodal Voltage Analysis Circuit

In the above circuit, node D is chosen as the reference node and the other three nodes are

assumed to have voltages, Va, Vb and Vc with respect to node D. For example;

As Va = 10v and Vc = 20v , Vb can be easily found by:

86

again is the same value of 0.286 amps, we found using Kirchhoff’s Circuit Law in the previous

tutorial.

From both Mesh and Nodal Analysis methods we have looked at so far, this is the simplest

method of solving this particular circuit. Generally, nodal voltage analysis is more appropriate

when there are a larger number of current sources around. The network is then defined as: [ I ]

= [ Y ] [ V ] where [ I ] are the driving current sources, [ V ] are the nodal voltages to be found

and [ Y ] is the admittance matrix of the network which operates on [ V ] to give [ I ].

Nodal Voltage Analysis Summary.

The basic procedure for solving Nodal Analysis equations is as follows:

1. Write down the current vectors, assuming currents into a node are positive. ie, a (N x

1) matrices for “N” independent nodes.

2. Write the admittance matrix [Y] of the network where:

Y11 = the total admittance of the first node.

Y22 = the total admittance of the second node.

RJK = the total admittance joining node J to node K.

3. For a network with “N” independent nodes, [Y] will be an (N x N) matrix and that Ynn will

be positive and Yjk will be negative or zero value.

4. The voltage vector will be (N x L) and will list the “N” voltages to be found.

We have now seen that a number of theorems exist that simplify the analysis of linear circuits.

In the next tutorial we will look at Thevenins Theorem which allows a network consisting of

linear resistors and sources to be represented by an equivalent circuit with a single voltage

source and a series resistance.

Mesh Current Analysis

Mesh Current Analysis is a technique used to find the currents circulating around a loop or

mesh with in any closed path of a circuit

While Kirchhoff´s Laws give us the basic method for analysing any complex electrical circuit,

there are different ways of improving upon this method by using Mesh Current

87

Analysis or Nodal Voltage Analysis that results in a lessening of the math’s involved and when

large networks are involved this reduction in maths can be a big advantage.

Mesh Current Analysis Circuit

One simple method of reducing the amount of math’s involved is to analyse the circuit using

Kirchhoff’s Current Law equations to determine the currents, I1 and I2 flowing in the two

resistors. Then there is no need to calculate the current I3 as its just the sum of I1 and I2. So

Kirchhoff’s second voltage law simply becomes:

Equation No 1 : 10 = 50I1 + 40I2

Equation No 2 : 20 = 40I1 + 60I2

therefore, one line of math’s calculation have been saved.

Mesh Current Analysis

An easier method of solving the above circuit is by using Mesh Current Analysis or Loop

Analysis which is also sometimes called Maxwell´s Circulating Currents method. Instead of

labelling the branch currents we need to label each “closed loop” with a circulating current.

As a general rule of thumb, only label inside loops in a clockwise direction with circulating

currents as the aim is to cover all the elements of the circuit at least once. Any required branch

current may be found from the appropriate loop or mesh currents as before using Kirchhoff´s

method.

For example: : i1 = I1 , i2 = -I2 and I3 = I1 – I2

We now write Kirchhoff’s voltage law equation in the same way as before to solve them but

the advantage of this method is that it ensures that the information obtained from the circuit

equations is the minimum required to solve the circuit as the information is more general and

can easily be put into a matrix form.

For example, consider the circuit from the previous section.

88

These equations can be solved quite quickly by using a single mesh impedance matrix Z. Each

element ON the principal diagonal will be “positive” and is the total impedance of each mesh.

Where as, each element OFF the principal diagonal will either be “zero” or “negative” and

represents the circuit element connecting all the appropriate meshes.

First we need to understand that when dealing with matrices, for the division of two matrices

it is the same as multiplying one matrix by the inverse of the other as shown.

having found the inverse of R, as V/R is the same as V x R-1, we can now use it to find the

two circulating currents.

89

Where:

[ V ] gives the total battery voltage for loop 1 and then loop 2

[ I ] states the names of the loop currents which we are trying to find

[ R ] is the resistance matrix

[ R-1 ] is the inverse of the [ R ] matrix

and this gives I1 as -0.143 Amps and I2 as -0.429 Amps

As : I3 = I1 – I2

The combined current of I3 is therefore given as : -0.143 – (-0.429) = 0.286 Amps

This is the same value of 0.286 amps current, we found previously in the Kirchhoffs circuit

law tutorial.

Mesh Current Analysis Summary

This “look-see” method of circuit analysis is probably the best of all the circuit analysis

methods with the basic procedure for solving Mesh Current Analysis equations is as follows:

1. Label all the internal loops with circulating currents. (I1, I2, …IL etc)

2. Write the [ L x 1 ] column matrix [ V ] giving the sum of all voltage sources in each loop.

3. Write the [ L x L ] matrix, [ R ] for all the resistances in the circuit as follows:

R11 = the total resistance in the first loop.

Rnn = the total resistance in the Nth loop.

RJK = the resistance which directly joins loop J to Loop K.

4. Write the matrix or vector equation [V] = [R] x [I] where [I] is the list of currents to be

found.