Subject Code: 23EEE104

Introduction to Electrical and Electronics Engineering

UNIT-I
Evaluation pattern
 Syllabus
• Introduction to Electrical Engineering, current and voltage sources, Resistance, Inductance,
and Capacitance

• Ohm’s law, Kirchhoff’s law, Energy and Power, Series parallel combination of R, L, and C
components

• Voltage Divider and Current Divider Rules, Superposition Theorem

• Network Analysis – Mesh and Node methods, Faraday’s Electro Magnetic Induction

• Magnetic Circuits-Self and Mutual Inductance,

• Generation of sinusoidal voltage, Instantaneous, Average and effective values of periodic

functions, Phasor representation.

• Introduction to 3-phase systems, Introduction to electric grids.

 Introduction to Electrical Engineering
• Electrical engineering is an engineering discipline concerned with the study, design, and application
of equipment, devices, and systems which use electricity, electronics, and electromagnetism.

• Examples of Electrical Devices: Transformers, generators, alternators, motors.

• Power Generation and Distribution: Working with power plants, transmission lines, and
transformers to generate and distribute electricity.

• Electrical Machines: Designing and maintaining motors, generators, and transformers.

• Industrial Automation: Involvement in the automation of industrial processes through the use of
electrical control systems.

• Renewable Energy Systems: Working on the generation of electricity from renewable sources like
solar, wind, and hydroelectric power.
• Electronics engineering : Primarily concerned with the design, development, and testing of
electronic circuits and devices.

• Examples of Electronics Devices: Computers, smartphones, consumer electronics, medical devices,

and communication systems.
• Consumer Electronics: Designing and developing products like smartphones, laptops, televisions, and home
appliances.

• Communication Systems: Developing communication devices and systems, such as mobile phones, satellite systems,
and wireless networks.

• Embedded Systems: Creating embedded systems used in various applications, from automotive electronics to medical
devices.

• Signal Processing: Working on systems that process and manipulate signals for applications in audio, video, and
telecommunications.

➢ Electrical Engineering deals with larger, power-related systems, while Electronics Engineering focuses on the
individual components within those systems (on smaller-scale, electronic components and systems).
 Electronic current

➢ Flow of free electrons (loosely bound electrons in the outmost orbit of its atoms) when
electrical pressure (voltage) applied.

✓ Strength of current (I) = flow charge (Q) per unit time

I = Q/t

➢ Units:
✓ Q – Coulomb (C)
✓ t – second (s)
✓ I – Ampere (A)
 Current and Voltage Sources
• Voltage sources and Current sources are both sources of Energy.

• Voltage source and current source both are electrical sources that provide electrical energy to
drive an electrical load.

• Current sources and voltage sources are fundamental components in electrical circuits that
provide the driving force for the flow of electric current.

• Current sources: A current source is a device that maintains a constant current flow through
its terminals, regardless of the voltage (varies) applied across them.

• Voltage sources: A voltage source is a device that maintains a constant voltage across its
terminals, regardless of the current (varies) drawn from them.
 Type of supplies

• Depending on the nature of the waveform, power supplies are classified as

1. Alternating current (AC)
2. Direct current (DC)

➢ DC is a current that remains constant with time.

➢ AC is a current that varies sinusoidally with time.
 Sources

Independent Dependent
source source

Voltage Voltage Current Current

Voltage Current controlled controlled controlled controlled
source source voltage current current voltage
source source source source
Classification of sources
Independent
sources

0 𝑓𝑜𝑟 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 𝑠𝑜𝑢𝑟𝑐𝑒

RS – internal resistance and ideal RS = ቊ
∞ 𝑓𝑜𝑟 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑠𝑜𝑢𝑟𝑐𝑒

VOUT = VL IOUT = IL
 Passive elements
• Passive elements are electronic components that do not introduce energy into a
circuit. Instead, they absorb, store, or dissipate electrical energy.

• These components do not require an external power source to function; they operate
solely based on their electrical properties.

• Passive elements do not amplify signals but play crucial roles in shaping and
controlling electrical circuits.

• Examples of passive elements include resistors, capacitors, inductors, and various

types of passive filters.
Passive components
 Active elements
• Active elements are electronic components that can generate, amplify, or control electrical
signals in a circuit.

• These components require an external source of energy (usually a power supply) to operate.

• Examples of active elements include transistors, operational amplifiers (op-amps), and

integrated circuits (ICs) like microcontrollers and microprocessors.
Linear and Nonlinear elements
• Linear elements are electrical components that follow the principles of linearity, meaning that their
behavior can be accurately described using linear mathematical equations (e.g., Ohm's Law).

• The voltage-current relationship for linear elements is characterized by a straight line when plotted on
a graph.

Eg: Resistors , Inductors (L) and Capacitors (C)((at low frequencies)

• Nonlinear elements are electrical components whose behavior cannot be accurately described by
linear mathematical equations.

• The voltage-current relationship for nonlinear elements does not follow a straight line when plotted
on a graph.

Eg: Diodes (in their active region), Transistors (in their active region), Nonlinear resistors
Unilateral and Bilateral Elements
• Unilateral elements" is a term used in electronics and electrical engineering to describe electronic components
or devices that exhibit different behaviors depending on the direction of current or voltage flow.

• These elements are not symmetrical with respect to electrical characteristics, and their behavior may vary
depending on whether the current or voltage is applied in a certain direction.

Eg: Diode as a Unilateral Element

• Bilateral elements" is a term used in electronics and electrical engineering to describe electronic components or
devices that exhibit the same electrical characteristics and behavior regardless of the direction of current or
voltage flow through them.

• In other words, bilateral elements are symmetrical in terms of their electrical properties, and their behavior
remains consistent whether the current or voltage is applied in one direction or the opposite direction.

• Eg: Resistance value and behavior of a resistor do not change based on the direction of current flow, inductors,
capacitors etc…
Time variant/invariant system

• If the element characteristics are independent of time, it is called a time-invariant, otherwise

time variant.

➢ Charging and Discharging Capacitor: A charging or discharging capacitor in an RC

circuit is a time-variant system.

As the capacitor charges or discharges, its voltage changes with time, making the system's
behavior dependent on the instantaneous time.

➢ Resistor in a DC Circuit: A simple resistor in a DC (direct current) circuit is a time-

invariant system. Ohm's Law (V = IR) applies consistently, and the resistor's behavior
remains the same over time.
Lumped and Distributed elements
• Lumped elements are components in a circuit that are assumed to have uniform and constant electrical
properties over their entire length.

• The lumped element approach is appropriate for low-frequency circuits where the physical size of the
components is small compared to the wavelength of the signals involved.

• Typical lumped elements are capacitors, resistors, and inductors

• Distributed elements on the other hand, are components in a circuit where the physical dimensions and
characteristics of the components significantly affect the behavior of signals as they propagate through
them.

• The distributed element approach is essential for high-frequency circuits, transmission lines, and RF (radio
frequency) designs because at high frequencies,

• For example, a transmission line has distributed parameters along its length and may extend for hundreds
of miles.
 Resistor
✓ Materials, in general, have a characteristic behavior of resisting the flow of electric charge.

✓ This physical property, or ability to resist current, is known as resistance and is represented by the
symbol R.

✓ The circuit element used to model the current-resisting behavior of a material is the resistor.

• Resistance: (R)

It is a property of a material that opposes the flow of electric current.

• Units: ohm’s “Ω”

• The resistance of any material with a uniform cross-sectional area A depends on A and its length ℓ,

• Resistance is directly proportional to the length of the material,

𝑅∝𝑙
• As the area of the cross section increases, electrons can move
freely.
• Resistance is inversely proportional to the area of the cross-
section.
R∝1/A

L is the length of the conductor in meters (m).

A is the cross-sectional area of the conductor in square meters (m2).
Contd…
R ∝ L/A

R = ρL/A

ρ=Resistivity (or) Specific Resistance

ρ = (RA)/L = (Ω∗m2)/m = Ω.m

• R is the electrical resistance in ohms (Ω).

• ρ (rho) is the resistivity of the material in ohm-meter (Ω⋅m).

• Resistance is a property of an object or electrical component, while resistivity is a property of the

material itself.

• Resistivity is a material property that characterizes how strongly a given material opposes the flow
of electric current.
Fixed Resistor symbol Variable Resistor symbol
Problem 1: Compute the resistivity of the Problem 2: The length and area of wire are given as
given material whose resistance is 2 Ω; area 0.2 m and 0.5 m2 respectively. The resistance of that
of cross-section and length are 25cm2 and 15 wire is 3 Ω, calculate the resistivity?
cm respectively?

(1cm=0.01m)
Calculating Resistance value
Series Circuit

Faulty Circuit
Healthy Circuit

Inferences from Series Circuit

▪ Current is same
▪ Voltage is shared among resistors
▪ Any fault cause entire circuit dead
Parallel Circuit

Healthy Circuit

Faulty Circuit
 Resistors in Series (Series Resistance)
➢ When a number of resistances are connected end to end across a source of supply, there will be only
one path for the current to flow as shown in Fig. The circuit is called a series circuit.

Fig. (a) Fig. (b)

➢ The voltage drops across the resistances are V1, V2 ,V3, and V4, respectively. Since the same
current is flowing through all the resistances, we can write
➢ Again, the total voltage, V applied is equal to the sum of the voltage drops across the resistances,
Thus we can write

➢ To find the value of equivalent resistance of a number of resistances connected in series we equate
the voltage, V of the two equivalent in units as shown in Fig. (a) and Fig. (b) as

➢ Thus, when resistances are connected in series, the total equivalent resistance appearing
across the supply can be taken as equal to the sum of the individual resistances.
 Resistors in Parallel (Parallel Resistance)
• When a number of resistors are connected in such a way that both the ends of individual resistors are
connected together and two terminals are brought out for connection to other parts of a circuit, then the
resistors are called connected in parallel as shown in Fig.

• Voltage V is connected across the three resistors R1, R2, R3 connected in parallel. The total
current drawn from the battery is I.
• This current gets divided into I1, I2, I3 such that I = I1 + I2 + I3. As voltage V is appearing
across each of these three resistors,

• Let the equivalent resistance of the three resistors connected in parallel across terminals A
and B be R as shown in Fig. (b). Then

• In general, if there are n resistors connected in parallel, the equivalent resistance R is

expressed as
Series–Parallel Circuits
3
4
 Capacitor
• A capacitor is a passive element designed to store energy in its electric field.
• Any two conducting surfaces separated by an insulating material (dielectric) is called a capacitor.

Circuit symbols for capacitors: (a) fixed capacitor, (b)

variable capacitor.

➢ In many practical applications, the plates may be

aluminum foil while the dielectric may be air,
ceramic, paper, or mica.
➢ Capacitors are used extensively in electronics,
A capacitor is typically constructed
as shown in Fig. communications, computers, and power systems.
• When a voltage source v is connected to the capacitor, as in Fig. the source deposits a positive charge
q on one plate and a negative charge −q on the other plate.
• The capacitor is said to store the electric charge.
• The amount of charge stored, represented by q, is directly proportional to the applied voltage V so
that

• where C is the constant of proportionality, is known

as the capacitance of the capacitor.
• The unit of capacitance is the farad (F)

Fig: Capacitor with applied voltage v.

• Capacitance: The ability of a capacitor to store charge is known as its capacitance. (or)
• Capacitance is the amount of charge stored per plate for a unit voltage difference in a capacitor.
𝑸
𝑪= 1 farad = 1 coulomb/volt.
𝑽

• Although the capacitance C of a capacitor is the ratio of the charge Q per plate to the applied voltage
v, it does not depend on Q or v.

• It depends on the physical dimensions of the capacitor

where
A is the surface area of each plate,
d is the distance between the plates, and
ϵ is the permittivity of the dielectric material between the plates.
Permittivity:
• Permittivity measures the ability of a material to store energy within the material.
• In simpler terms, permittivity indicates how much resistance a material offers to the formation of an
electric field within it.
• often denoted by the symbol “ε”,

Three factors determine the value of the capacitance:

1. The surface area of the plates—the larger the area, the greater the capacitance.
2. The spacing between the plates—the smaller the spacing, the greater the capacitance.
3. The permittivity of the material—the higher the permittivity, the greater the capacitance.
• Permittivity is a fundamental property of materials in electrostatics and electromagnetism. There are

two key types:

1. Absolute permittivity (ε): This refers to the permittivity of a material in comparison to free space.

2. Relative permittivity (εr): also called the dielectric constant: This is the ratio of the permittivity of a

material to the permittivity of vacuum (free space).

Mathematically:
𝜺
𝜺𝒓 =
𝜺𝟎

• where ε0 is the permittivity of free space, approximately 8.854 *10-12 F/m (farads per meter).

• Permittivity plays an essential role in the behavior of capacitors and in understanding how materials

influence electric fields.

To obtain the current-voltage relationship of the capacitor
For a nonlinear capacitor
𝑑𝑉
i= C
𝑑𝑡

The voltage-current relation of the capacitor can be

obtained by integrating both sides

𝑑𝑉
Capacitors that satisfy i= C are said to be linear.
𝑑𝑡
 Energy stored in capacitor:
• The instantaneous power delivered to the capacitor is

• Let us consider that ‘V’ voltage is applied across the capacitor. At this instant, ‘W’ joules of work will
be done in transferring 1C of charge from one plate to another.

The energy stored in the capacitor is therefore

We note that v(−∞) = 0, because the capacitor was uncharged at t = −∞. Thus,
Example 1: A parallel plate capacitor has square plates of side 5 cm and is separated by a
distance of 1 mm. (a) Calculate the capacitance of this capacitor. (b) If a 10 V battery is
connected to the capacitor, what is the charge stored in any one of the plates? (The value of
εo = 8.85 x 10-12 Nm2 C-2)
Solution
(a) The capacitance of the capacitor is

C= 221.2 ×10−13 F
C = 22.12 ×10−12 F = 22 .12 pF
(b) The charge stored in any one of the plates is Q = CV, Then
= 22.12 ×10−12 ×10 = 221.2 ×10−12 C = 221.2 pC
45
Example 2:
(a) Calculate the charge stored on a 3-pF capacitor with 20 V across it.
(b) Find the energy stored in the capacitor.
Solution:

(a) Since q = Cv,

(b) The energy stored is

Series Capacitors

• The equivalent capacitance of series-connected

capacitors is the reciprocal of the sum of the
reciprocals of the individual capacitances.
Parallel Capacitors

• The equivalent capacitance of N parallel-connected capacitors is the sum of the individual

capacitances.
1. Find the equivalent capacitance seen between terminals a and b
of the circuit in Fig.

2. Find the equivalent

capacitance seen at the
terminals of the circuit.
Solution:
• The 20-μF and 5-μF capacitors are in series; their equivalent
capacitance is

• This 4-μF capacitor is in parallel with the 6-μF and 20-μF

capacitors; their combined capacitance is

4 + 6 + 20 = 30 μF
Answer: 40 μF.
• This 30-μF capacitor is in series with the 60-μF capacitor.
Hence, the equivalent capacitance for the entire circuit is
 Inductor
➢ An inductor is a passive component used in most power electronic circuits to store energy in the
form of magnetic energy when electricity is applied to it.

➢ One of the key properties of an inductor is that it opposes any change in the amount of current
flowing through it.

➢ Inductance : “The property of coil that opposes any change in the amount of current flowing through
it is called as Inductance”.

➢ Inductance is a result of the induced magnetic field on the coil.

➢ Whenever the current across the inductor changes, it either acquires charge or loses the charge in
order to equalize the current passing through it.

➢ The inductor is also called a choke, a reactor or just a coil. An inductor is a wire loop or coil in
its most basic form.)
• If current is allowed to pass through an inductor, it is found that the voltage across the
inductor is directly proportional to rate of change of the current.

• Flux linkage depends on the amount of current flowing through the coil.

• ∴ψ∝i
• ψ = Li [L is the constant of proportionality called the
Inductance, L=Inductance of coil]

• The SI unit of inductance is henry (H)

• The inductance of an inductor depends on its physical dimension and construction.

➢ Permeability is a property of a material It

essentially measures the material's ability to allow
where
magnetic lines of force (magnetic flux) to pass
• N is the number of turns,
through it.
• ℓ is the length,
➢ Symbol μ and is measured in Henries per meter
• A is the cross-sectional (H/m) or Newton per ampere squared (N/A²).
• area, and ➢ Different materials have different permeabilities
• μ is the permeability of the core.
• The current-voltage relationship is obtained

Integrating gives

• The current through an inductor cannot change instantaneously

Energy stored in inductor:
The power delivered to the inductor is

The energy stored is

Since i (−∞) = 0,
 Properties of inductor:

• Property 1: Inductors store kinetic energy in the form of magnetic energy. The formula for
energy stored in the magnetic field is equal to E = (½)LI2, where L is the inductance, and I is
the current.

• Property 2: Inductors allow only direct current (DC) to pass through it while blocking the
alternating current (AC). These types of inductors are called chokes.

• Property 3: Inductors consume power from the power source.

• Property 4: In a pure inductive circuit, the current lags behind voltage by 900.

• Property 5: Inductors oppose current change for alternating current.

Inductors can be used for two primary functions:

1.To control signals.

2.To store energy.

➢ Controlling Signals

• A choke is a type of inductor that is used mainly for blocking high-frequency alternating current
(AC) in an electrical circuit.

• As the function of this inductor is to restrict the changes in current, it is called a choke.

Storing Energy

• Inductor stores energy in the form of magnetic energy. Coils can store electrical energy in the form
of magnetic energy, using the property that an electric current flowing through a coil produces a
magnetic field, which in turn, produces an electric current.
Series Inductors
Consider a series connection of N inductors

The inductors have the same current through them

➢ The equivalent inductance of series-

connected inductors is the sum of the
individual inductances.
Parallel Inductors

• Consider a parallel connection of N inductors

• The inductors have the same voltage across them.

• The equivalent inductance of parallel inductors is the

reciprocal of the sum of the reciprocals of the
individual inductances.
Problem: Find the equivalent inductance of the circuit shown in Fig.

Solution:
➢ The 10H, 12H, and 20H inductors are in series; thus,
combining them gives a 42-H inductance.
➢ This 42-H inductor is in parallel with the 7H inductor so
that they are combined, to give

This 6H inductor is in series with the 4H and 8H inductors.

Hence,
Leq = 4 + 6 + 8 = 18 H
Important characteristics of the basic elements
Concept of current flow in conductor
 Ohm’s Law
• This law gives relationship between the potential differences (V), the current (I) and the
resistance (R) of a d.c. circuit.

• Dr. Ohm in 1827 discovered a law called Ohm`s law.

• It states that, “The current flowing through the electric circuit is directly proportional to the
potential difference across the circuit and inversely proportional to the resistance of the
circuit, provided the temperature remains constant”.

• Where

• I is the current flowing in amperes

• The V is the voltage applied and
• R is the resistance of the conductor.
• The unit of potential difference is defined in such a way that the constant of proportionality is
unity.
𝑽
𝑰= 𝒂𝒎𝒑𝒆𝒓𝒆𝒔 𝑽 = 𝑰𝑹 volts
𝑹

𝑽
= 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 𝑹 𝑂ℎ𝑚𝑠
𝑰

• The Ohm’s law can be defined as, “ The ratio of potential differences (V) between any two points of a
conductor to the current (I) flowing between them is constant, provided that the temperature of the
conductor remains constant.”
Limitations of Ohm’s law:

• This is not applicable to nonlinear devices such as diodes, Zener diodes, Voltage regulators, etc.

• It does not hold good for non-metallic conductors such as silicon carbide. The law for such conductors
is given by
• 𝑉 = 𝐾𝐼 𝑚 𝑤ℎ𝑒𝑟𝑒 𝐾, 𝑚 𝑎𝑟𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠.
Examples
Current?
Voltage?

Resistance?
 Kirchhoff’s Laws: KVL, KCL
• Kirchhoff’s laws are two in number, are particularly useful (a) in determining the
equivalent resistance of a complicated network of conductors and (b) for calculating the
currents flowing in the various conductors. The two laws are,
(i) Kirchhoff’s Current Law (KCL):

• In any electrical network, the algebraic sum of the currents meeting at a point (or junction)
is zero.
• It simply means that the total current leaving a junction is equal to the total current
entering that junction.
• It is obviously true because there is no accumulation of charge at the junction of the
network.
• Consider the case of few conductors meeting at a point A as in fig.. Some conductors
have currents leading to point A, whereas some have currents leading away from point A.

• Assuming the incoming currents to be positive and the outgoing currents negative, we have

𝐼1 + −𝐼2 + −𝐼3 + +𝐼4 + (−𝐼5)= 0

or
𝐼1 + 𝐼4 − 𝐼2 − 𝐼3 − 𝐼5 = 0

𝐼1 + 𝐼4 = 𝐼2 + 𝐼3 + 𝐼5
sum of incoming currents = sum of outgoing currents
(ii) Kirchhoff’s Mesh Law or Voltage Law ( KVL):

• In any closed electrical circuit or mesh, the algebraic sum of

all the electromotive forces (e.m.f’s) or voltage drops in
resistors is equal to zero.

In any closed circuit or mesh,

• Algebraic sum of e.m.f’s + Algebraic sum of voltage drops = 0

• Below figure shows the circuit which explains KVL, here

VS is the source voltage V1 is the voltage drop across the
resistor R1 and V2 is the voltage drop across the resistor R2.

According to KVL

𝑉𝑠 + (−𝑉1) + (−𝑉2) = 0
𝑉𝑠 = 𝑉1 + 𝑉2
 Example 1
If R1 = 2Ω, R2 = 4Ω, R3 = 6Ω, determine the electric current that flows in the circuit below.

– IR1 + E1 – IR2 – IR3 – E2 = 0

Substituting the values in the equation, we get
–2I + 10 – 4I – 6I – 5 = 0
-12I + 5 = 0
I = -5/-12
I = 0.416 A 70
Calculate applying Kirchhoff ’s laws the current flowing through the 8  resistor in the circuit
shown in Fig.1

Fig.1 Fig.2
By observing the given circuit we see that nodes A, B, C are at the same potential and they
can be joined together so that the circuit will be like shown in Fig.2
• In the loop EAFE, current I1 will flow. No current from this loop will flow to the other two loops.
Current flowing from E to A is to be the same as the current flowing from A to F.

• The distribution of currents in loop GDAG and HDAH have been shown. By applying KVL in these
loops we write:

for loop GDAG

Solving eqs. (i) and (ii)

Current through the 8Ω resistor = I2 + I3

for loop HDAH
Nodes, Branches, and Loops
• A branch represents a single element such as a voltage source or a resistor.

• A node is the point of connection between two or more branches.

• A loop is any closed path in a circuit.

 Work, Power, Energy
➢ Work

• When a force is applied to a body causing it to move, and if a displacement, d is caused in the
direction of the force, then

Work done = Force × Distance

W=F×d

• If force is in Newtons and d is in meters, then work done is expressed in Newton–meter which
is called Joules.
Power
• Power is the rate at which work is done, i.e., rate of doing work. Thus,

• The unit of power is Joules/second which is also called Watt.

• We have that electrical potential, V is expressed as

• Thus, in a circuit if I is the current flowing, and V is the applied voltage across the terminals,
power, P is expressed as

➢ The electrical power can be expressed as

 ➢ Energy
• Energy is defined as the capacity for doing work.

• The total work done in an electrical circuit is called electrical energy.

• When a voltage, V is applied, the charge, Q will flow so that

Electrical energy = Power × Time

If electrical power is expressed is kW and time in hour, then

Electrical energy = kWh
 Voltage Divider Rule
• For easy calculation of voltage drop across resistors in a series circuit, a voltage divider rule is used
which is illustrated in Fig.

Thus the voltage divider rule states that voltage drop across any
resistor in a series circuit is proportional to the ratio of its
resistance to the total resistance of the series circuit.
 Current Divider Rule
• Current divider rule is used in parallel circuits to find the branch currents if the total current is
known. To illustrate, this rule is applied to two parallel branches as in Fig.

VAB

Method I

or,
 Method II
Step 1: Find out the total resistance of the circuit.
R1 R 2
R1 R 2 VAB I ∗ RT R1 + R 2 R1
RT = I2 = = =I∗ =I∗
R1 + R 2 R2 R2 R2 R1 + R 2

Step 2: Calculate the total current pass through the

circuit. Thus, in a parallel circuit of two
VAB
I= VAB = I ∗ R T resistances, current through one branch is
RT
equal to line current multiplied by the ratio
Step 3: Calculate the current passing through each
of resistance of the other branch divided by
resistor.
the total resistance.
R1 R 2
VAB I ∗ RT R1 + R 2 R2
I1 = = =I∗ =I∗
R1 R1 R1 R1 + R 2
Find the values of different voltages that can be obtained from 25V source with the help of
voltage divider circuit of fig.

• Solution: Total circuit resistance, 𝑅𝑇 = 𝑅1 + 𝑅2 + 𝑅3 = 1 + 8.2 + 3.3 = 12.5 𝑘Ω

𝑅1 1
• Voltage drop across 𝑅 1, 𝑉1 = × 𝑉𝑆 = × 25 = 2 𝑉
𝑅𝑇 12.5
• Voltage at point B, 𝑉𝐵 = 25 − 2 = 23 𝑉
𝑅2 8.2
• Voltage drop across 𝑅 2, 𝑉2 = × 𝑉𝑆 = × 25 = 16.4 𝑉
𝑅𝑇 12.5
• Voltage at point C, 𝑉𝐶 = 𝑉𝐵 − 𝑉2 = 23 − 16.4 = 6.6 𝑉
• The different available load voltages are :
• 𝑉𝐴𝐵 = 𝑉𝐴 − 𝑉𝐵 = 25 − 23 = 2𝑉;
• 𝑉𝐴𝐶 = 𝑉𝐴 − 𝑉𝐶 = 25 − 6.6 = 18.4 𝑉;
• 𝑉𝐵𝐶 = 𝑉𝐵 − 𝑉𝐶 = 23 − 6.6 = 16.4 𝑉; 𝑉𝐴𝐷 = 25 𝑉
• 𝑉𝐶𝐷 = 𝑉𝐶 − 𝑉𝐷 = 6.6 − 0 = 6.6 𝑉; 𝑉𝐵𝐷 = 𝑉𝐵 − 𝑉𝐷 = 23 − 0 = 23 𝑉;
Figure shows the voltage divider circuit. Find (i) the current drawn from the supply, (ii) voltage
across the load RL, (iii) the current fed to RL and (iv) the current in the tapped portion of the
divider.
• Solution: It is a loaded voltage divider.

• (i) 𝑅𝐵𝐶 = 120Ω ∥ 300Ω = 120×300 = 85.71Ω

120+300
𝑅 𝐴𝐵 80
• 𝑉𝐴𝐵 = × 𝑉𝑆 = × 200 = 96.55 𝑉
𝑅 𝐴𝐵 +𝑅 𝐵𝐶 80+85.71

• The current I drawn from the supply is 𝐼 = 𝑉 𝐴𝐵 = 96.55 = 1.21 𝐴

𝑅 𝐴𝐵 80
𝑅 𝐵𝐶 85.71
• (ii) 𝑉𝐵𝐶 = × 𝑉𝑆 = × 200 = 103.45 𝑉
𝑅 𝐴𝐵 +𝑅 𝐵𝐶 80+85.71

• (iii) Current fed to load 𝐼𝐿 = 𝑉 𝐵𝐶 = 103.45 = 0.35 𝐴

𝑅𝐿 300
• (iv) Current in the tapped portion of the divider is 𝐼𝐵𝐶 = 𝐼 − 𝐼𝐿 = 1.21 − 0.35 = 0.86 𝐴
Problem: Find the branch currents for fig. 14 using the current divider rule for parallel
conductance.

• Solution:
• 𝐺𝑃 = 𝐺1 + 𝐺2 + 𝐺3 = 0.5 + 0.3 + 0.2 = 1

• 𝐼1 = 𝐼 × 𝐺1 = 4 × 0.5 = 2 𝐴
𝐺𝑃 1

• 𝐼2 = 𝐼 × 𝐺2 = 4 × 0.3 = 1.2 𝐴
𝐺𝑃 1

• 𝐼3 = 𝐼 × 𝐺3 = 4 × 0.2 = 0.8 𝐴
𝐺𝑃 1
 Network Analysis methods
• There are a variety of techniques, all based two laws (KVL, KCL) that can simplify circuit analysis.

Mesh analysis

Nodal analysis

➢ Mesh Analysis (Mesh Current method)

• A mesh is a smallest loop in a network. KVL is applied to each mesh in terms of mesh currents instead
of branch currents.

• As a convention, mesh currents are assumed to be flowing in the clockwise direction without
branching out at the junctions.

• Applying KVL, the voltage equations are framed.

• By knowing the mesh currents, the branch currents can be determined.

• Using the mesh current method calculate the current flowing through the resistors in the
circuit shown in Fig
Applying KVL in mesh II

Adding eqs. (i) and (ii)

Applying KVL in mesh I

Current through the 6 Ω resistor is (I1 − I2)

which is equal to 2/33 A.
Nodal Analysis

• This technique of circuit solution, also known as the Node voltage method, is based on the application
of KCL at each junction (node) of the circuit, to find the node voltages.

• This method of circuit analysis is suitable where a network has a number of loops, and hence a large
number of simultaneous equations are to be solved.

➢ Steps to Determine Node Voltages:

• Step 1: Give a reference node in the network.

• Step 2: Represent the currents in all the branches and represent voltage at each node with respect to
the reference node.

• Step 3: Write each branch current using the represented node voltages (apply ohm’s law).

• Step 4: Apply KCL at the appropriate nodes.

• Step 5: Calculate branch currents or node voltages using step 3 & 4.

For the circuit shown in Fig. 1 determine the voltages at nodes B and C and calculate the current
through the 8 resistor.

Solution:
• We will take one reference node at zero potential.
• Generally, the node at which maximum branches are meeting is taken as the reference node.
• Let R is the reference node as shown in Fig.2. The reference node will be called ground node
or zero potential node.
 To find I3, we assume potential at point
k as VK. We can write, VK + 3 = VB

• Points F, G, R, H, I are at zero reference potential.

• Let us now assign potential at all nodes with respect to
Applying KCL at node B,
the reference node.
• Let VD, VB, VC, VE are the potentials at points D, B, C,
and E, respectively.
• Let us also assume unknown currents II, I2, I3, I4, and I5
flowing through the branches.
 Solving eqs. (i) and (ii), we get

Applying KCL at node C,

and current in 8 resistor,
Problem: A network with three meshes has been Problem: Use nodal analysis to determine the
shown in below. Applying Maxwell’s mesh current flowing through the various branches in
current method determine the value of the the circuit shown in Figure. All resistances
unknown voltage, V for which the mesh current, shown are in Ohms.
I1 will be zero. Ans: V = 48 V
Using the Node voltage method calculate the voltages V1 and V2 in Fig 1. and hence calculate the
currents in the 8 resistor.

Solution: Step 3: Apply Kirchhoff ’s current law at each node.

At node 1:
Step 1 Reference node shown. Voltages V1 and V2
assigned.
Step 2 Assign currents in each connection to each node
 Multiply each term by 30:

Equation (c) - equation (b) gives

At node 2:

From (a)
Multiply each term by 120:

Hence the current in the 8 resistor is

Step 4: Solve for V1 and V2.
 Superposition Theorem
• The superposition theorem states that in a linear network containing more than one source, the
current flowing in any branch is the algebraic sum of currents that would have been produced
by each source taken separately. All the other sources replaced by their respective internal
resistances. In case the internal resistance of a source is not provided, the voltage sources will
be short circuited and current sources will be open circuited.

Example: Using the superposition theorem find the value of current, IBD in the circuit.
➢ Statement: The superposition principle states that the voltage across (or current through) an element
in a linear circuit is the algebraic sum of the voltages across (or currents through) that element due to
each independent source acting alone.

➢ Steps to apply superposition principle:

1. Turn off all independent sources except one source. Find the output (voltage or current) due
to that active source using nodal or mesh analysis.

2. Repeat step 1 for each of the other independent sources.

3. Find the total contribution by adding algebraically all the contributions due to the independent
sources.

Turn off voltage sources = short voltage sources; make it equal to zero voltage

Turn off current sources = open current sources; make it equal to zero current
Solution:

• We shall consider each source separately and calculate the current flowing through the branch BD.
First the 24 V source is taken by short circuiting the 12 V source as shown in Fig.

This I = A 8/3 gets divided into two parts as I1 and I2. Current through the resistor across BD is
I1. To find I1 we can use the current division rule as
➢ Now, consider the 12V source and short circuit the 24V source as shown in Fig. The current
supplied by the 12V source is calculated as

➢ The total current I due to 12 V supply has been calculated as 4/3A. This current gets divided
into I1 and I2 as has been shown in Fig. (c). Current I1 is calculated using the current division
rule as
➢ To determine the current flowing through the resistor across BD, the combined effect
of the two voltage sources will be taken. Therefore,
 source transformation
• A source transformation is the process of replacing a voltage source Vs in series with a
resistor R by a current source Is is in parallel with a resistor R, or vice versa.

• Source transformation is another tool for simplifying circuits. Basic to these tools is the
concept of equivalence.

• We recall that an equivalent circuit is one whose v-i characteristics are identical with the
original circuit.

• Source transformation does not affect the remaining part of the circuit.
Voltage source into current source

Current source into voltage source

Rse = Rsh
 Example problems
➢ Convert a voltage source of 20 volts with internal resistance of 5 Ω into an equivalent current source.

➢ Convert a current source of 100 A with internal resistance of 10 Ω into an equivalent voltage source.
Key differences between resistance and reactance:

Feature Resistance (R) Reactance (X)

Opposition to the flow of both AC and Opposition to the flow of AC only (due
Nature
DC. to phase shift).

Units Ohms (Ω) Ohms (Ω)

Stores energy in magnetic or electric

Energy Loss Causes power dissipation as heat.
fields (no power loss).

Depends on the frequency of AC

Frequency Dependence Independent of frequency. (increases for inductors, decreases for
capacitors).

No phase shift between voltage and Causes phase shift between voltage and
Phase Shift
current. current.
 Series R, L,C circuit
• A series RLC Circuit is an electrical circuit that consists of three components resistor (R), inductor
(L), and capacitor (C) connected in series.
• The circuit is driven by an alternating current (AC) voltage source.
• Each component behaves differently with AC.

➢ Inductor: Opposes any change in

the amount of current flowing
through it.

➢ Capacitor doesn’t allow the sudden

change in voltage.
 Impedance (Z)
• Impedance (Z) is the total opposition that a circuit presents to the flow of alternating current
(AC)

• Impedance determines how much current will flow in an AC circuit for a given applied
voltage and is measured in ohms (Ω).

• It is the AC equivalent of resistance of DC

• It incorporating both the effects of resistance and reactance.

• Reactance: Is the opposition that inductors and capacitors provide to alternating current
(AC), due to their phase-shifting properties.

• Resistance dissipates energy, while reactance stores and releases energy without dissipating
it.
➢ Reactance is denoted by X and can be of two types:
Capacitive Reactance (𝑋𝐶): Opposition offered by a capacitor to AC. It decreases as the frequency
increases. 1
𝑋𝑐 =
2𝜋𝑓𝐶

Inductive Reactance (𝑋𝐿): Opposition offered by an inductor to AC. It increases as the frequency
increases.
𝑋𝐿=2𝜋𝑓𝐿

▪ The circuit draws a current I. Due to flow of current I, there are voltage drops across R, L, and C.
Which are given by
(i) drop across resistance R is VR = IR
(ii) drop across inductance L is VL = IXL
(iii) drop across capacitance C is VC = IXC
➢ Impedance in different components:

• Resistor (R): The impedance is purely real, equal to its resistance:

• Inductor (L): The impedance is purely imaginary and increases with frequency

• Capacitor (C): The impedance is also imaginary but decreases with increasing frequency
Instantaneous voltages for a Series RLC Circuit:

➢ 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.
Phasor diagram of series RLC Circuit

Voltage triangle for a series RLC circuit

In a series RLC circuit the total impedance is:

Phase,

➢ Impedance affects both the magnitude and phase of the current and voltage in an AC circuit.
➢ It is frequency-dependent due to the reactance from inductors and capacitors.
 Parallel RLC circuit
➢ A Parallel RLC Circuit consists of a resistor (R),
inductor (L), and capacitor (C) connected in parallel
across an alternating current (AC) voltage source.

➢ In the parallel configuration, the voltage across

each component is the same, but the current
through each component varies depending on its
impedance.

➢ Admittance (Y): Is a measure of how easily a circuit or device will allow a current to flow.
It is the reciprocal of impedance.
 Phasor diagram of parallel RLC Circuit

Current Triangle for a Parallel RLC Circuit

The magnitude of the total impedance Z
in a parallel RLC circuit is:
 Electromagnetism and Electromagnetic Induction

• When current, I applied to the coil, there was a

magnetic field around a current-carrying conductor.
• Lines of force in the form of concentric circles existed
on a perpendicular plane around a current-carrying
conductor.
• This meant, magnetism could be created by electric
current.
• It was also observed that the direction of lines of force
got changed when the direction of current flowing
through the conductor was changed.

Coil convert electric energy into magnetic field

(a) Right-hand-grip rule applied to determine direction of flux produced by a current-carrying coil
 Faraday’s Laws of Electro-magnetic Induction
• Whenever there a is change in the magnetic flux linkage by a coil, EMF is induced in the coil.

• The magnitude of the EMF induced is proportional to the rate of change of flux linkages.

• Faraday’s laws of electromagnetic induction are stated as:

• First law: EMF is induced in a coil whenever magnetic field linking that coil is changed.

• Second law: The magnitude of the induced EMF is proportional to the rate of change of flux linkage.

The rate of change of flux linkage is expressed as

N is the number of turns of the coil

Thus, the induced EMF, e is expressed as

 The minus sign is introduced in accordance with Lenz’s law
Lenz’s law:
• This law states that the induced EMF due to change of flux linkage by a coil will produce a current
in the coil in such a direction that it will produce a magnetic field which will oppose the cause, that
is the change in flux linkage.

(a)Faraday’s experiment on electromagnetic (b) Determination of the direction of

induction current produced in the coil

The direction of current flowing through the coil can be determined by applying the right-hand-grip rule.
The similarity between Magnetic and Electric Circuits
 Dynamically Induced EMF and Statically Induced EMF

➢ Dynamically Induced EMF

• When EMF is induced in a coil or conductor by virtue of movement of either the

conductor or the magnetic field, the EMF is called dynamically induced EMF.

➢ Statically Induced EMF

• When EMF is induced in a stationary coil by changing its flux linkage due to change in
current flow through the coil, such emf is called statically induced EMF.
➢ Self Inductance or Self-induced EMF
• The EMF induced in a coil due to change in flux linkage when a changing current flows through the
coil is called self-induced EMF.
where
B is the magnetic flux density in Wb/m2.
μ is the permeability of the core material;
H Magnetic field strength
l is the length of flux path;
A is the area of the coil.

Permeability is a property of a material it essentially measures the

material's ability to allow magnetic lines of force (magnetic flux) to
pass through it.
➢ L is called the coefficient of self inductance or simply self
inductance of the coil.
➢ Inductance of a coil is, therefore, dependent upon the
permeability of the core material.
➢ Mutual inductance or Mutually induced EMF
• When a second coil is brought near a coil producing changing flux, EMF will be induced in the
second coil due to change in current in the first coil. This is called mutually induced EMF.
• The magnitude of the induced EMF will depend upon the rate of change of flux linkage and the
number of turns of the individual coils. The induced EMF in the two coils, e1 and e2 will be

N1 and N2 are the number of turns of coil 1 and coil 2

▪ Consider two coils having N1 and N2 number of turns placed near each other as shown in Fig.
▪ Let i1, flow through coil 1. The flux produced by i1 in N1 is 1.
▪ Since coil 2 is placed near coil 1, a part of the flux produced by coil 1 will be linked by coil 2.
▪ Let flux 2 linked by coil 2 is 2. (where K1<1).
▪ If magnetic coupling between the two coils is very tight, i.e., very good, the whole flux
produced by coil 1 will link the coil 2, in which case the coefficient of the coupling K1 will be
1. The induced EMF in coil 2 is e2.
 where,

is called the mutual inductance of the two coils.

Similarly, if we calculate the induced EMF in coil 1, due to change in current in coil 2, we
can find the induced EMF e1 in coil 1 as
Example 1.
Consider a solenoid with 500 turns which are wound on an iron core whose relative permeability is 800.
40 cm is the length of the solenoid, while 3 cm is the radius. The change in current is from 0 to 3 A.
Calculate the average emf induced for this change in the current for a time of 0.4 seconds.

Solution:
Self-inductance is given as
Given:
L = μN2A/l = μ0μrN2𝜋r2/l
No.of turns, N = 500 turns
Substituting the values we get
Relative permeability, μr = 800
(4)(3.14)(10-7)(800)(5002)(3.14)(3×10-2)2/0.4
Length, l = 40 cm = 0.4 m
L = 1.77 H
Radius, r = 3 cm = 0.03 m
Magnitude of induced emf, ε = L di/dt = 1.77×3/0.4
Change in current, di = 3 – 0 = 3 A
ε = 13.275 V
Change in time, dt = 0.4 sec

123
 Introduction to AC circuits
• The circuit that is excited using alternating source is called an AC Circuit. The alternating
current (AC) is used for domestic and industrial purposes.

• In an AC circuit, the value of the magnitude and the direction of current and voltages is not
constant, it changes at a regular interval of time.

• In AC Circuit, the current and voltages are represented by magnitude and direction.

• The alternating quantity may or may not be in phase with each other depending upon the
various parameters of the circuit like resistance, inductance, and capacitance.

• The sinusoidal alternating quantities are voltage and current which varies according to the
sine of angle θ.
 Terminology
• Amplitude: The maximum positive or negative value attained by an alternating quantity in
one complete cycle is called Amplitude or peak value or maximum value. The maximum
value of voltage and current is represented by Em or Vm and Im respectively.

• Alternation: One-half cycle is termed as alternation. An alternation span is of 180 degrees

electrical.

• Cycle: When one set of positive and negative values completes by an alternating quantity or
it goes through 360 degrees electrical, it is said to have one complete cycle.

• Instantaneous value: The value of voltage or current at any instant of time is called an
instantaneous value. It is denoted by (i or e).
• Frequency: The number of cycles made per second by an alternating quantity is called
frequency. It is measured in cycle per second (c/s) or hertz (Hz) and is denoted by (f).

• Time Period: The time taken in seconds by a voltage or a current to complete one cycle is
called time period. It is denoted by (T).

• Waveform: The shape obtained by plotting the instantaneous values of an alternating

quantity such as voltage and current along the y-axis and the time (t) or angle (θ=wt) along
the x-axis is called a waveform.
 Introduction to Phasor diagram
• Phasor Diagrams are a graphical way of representing the magnitude and directional
relationship between two or more alternating quantities.

• Basically, a rotating vector, simply called a “Phasor” is a scaled line whose length represents
an AC quantity that has both magnitude (“peak amplitude”) and direction (“phase”) which is
“frozen” at some point in time.

• A phasor is a vector that has an arrow head at one end which signifies partly the maximum
value of the vector quantity ( V or I ) and partly the end of the vector that rotates.
Contd..
• Anti-clockwise rotation of the vector is considered to be a positive rotation. Likewise, a
clockwise rotation is considered to be a negative rotation.

• A complete sine wave can be constructed by a single vector rotating at an angular velocity
of ω = 2πƒ, where ƒ is the frequency of the waveform. Then a Phasor is a quantity that has
both “Magnitude” and “Direction”.

• Generally, when constructing a phasor diagram, angular velocity of a sine wave is always
assumed to be: ω in rad/sec.
Phasor diagram of Sinusoidal waveform:
• As the single vector rotates in an anti-clockwise direction, its tip at point A will rotate one
complete revolution of 360o or 2π representing one complete cycle.

• When the vector is horizontal the tip of the vector represents the angles at 0o, 180o and at
360o.

• Likewise, when the tip of the vector is vertical it represents the positive peak value, ( +Am )
at 90o or π/2 and the negative peak value, ( -Am ) at 270o or 3π/2.

• Then the time axis of the waveform represents the angle either in degrees or radians through
which the phasor has moved.