# BEEE 1st Year (Unit 1)

## Course Syllabus and Suggestive Readings

__Unit.1: ____Electrical and Magnetic Circuits__

__Unit.1:__

__Electrical and Magnetic Circuits__

__Chapter:1.DC & AC Circuits__

Introduction to DC and AC circuits, Active and passive two terminal elements, Ideal and Practical sources, Ohms law, Voltage-Current relations for resistor, inductor, capacitor, Kirchhoff’s laws, Analysis of R-L, R-C and R-L-C series circuits.

__Chapter:2.Magnetic Circuits and Transformers__

Magnetic effects of electric current, Law of Electromagnetic Induction, Self-Inductance, Mutual Inductance, Single Phase Transformer: Construction, Working principle, Efficiency.

## Unit 1, 2, and 3 Handwritten Notes PDF

## Lecture Topic 1.1.1

__Introduction to DC & AC CIRCUITS__

**1. Introduction: **As we know there are two types of supply which are alternating current and direct current. Both the currents are used in their suitable applications. AC supply mostly is preferred for our daily home appliances like washing machines, refrigerator, drilling machines, transformers while DC supply is used for laptops, cell phones, battery charging applications. Both the supplies have their advantages and disadvantages.

**AC supply:** It is basically defined as a supply in which magnitude and directions both vary with respect to time.

**DC supply:** It is defined as a supply in which magnitude and directions both are constant at any instant of time.

**2. Basic electrical elements:**

**Resistance: **It is the property of resistor which opposes the flow of charge or free electrons. It is represented by R and its unit is in ohm.

**Inductance: **It is property in which property of inductor by virtue of which it opposes any change of magnitude and direction of electric current passing through conductor. It stores energy in the form of magnetic field. It is represented by L and its unit is henry.

**Capacitance:** It is the property of two plate capacitor which stores energy in the form of electric field. It is represented by C and its unit is in farad.

**3. Types of elements: **On the basis of different parameters there are different types of elements such as:

- Active and passive
- Linear and non-linear
- Unilateral and bi-lateral

**Active and passive elements: **Active elements are those elements in which they have their own source of energy. They do not require any external source of energy.

exp. Voltage source, current source

Passive elements are those which need external source of energy.

exp. resistance, inductance and capacitance

**Linear and non-linear elements:** Linear elements are those elements in which they show linear characteristics.

They follow ohm’s laws.

exp. Resistance, inductance, capacitance

Non-linear elements are those which do not follow ohm’s law hence these are called non-linear elements.

exp. Diode, transistor

**Unilateral and bilateral elements:** Unilateral elements are those in which conduction of current is in one direction.

exp. Diode, rectifier

Bilateral element are those in which conduction of current is in both direction.

exp. Voltage source, current source, resistance, inductance, capacitance.

**4.Classification of sources:**

Fig.1. Classification of sources

**Ideal and practical voltage source: **An ideal voltage source is that source in which any variation in current does not impact on voltage source otherwise it is called practical voltage source.

Fig.2. Ideal and practical voltage source

**Ideal and practical current source: **An ideal current source is defined as a source in which any variation in voltage does not impact on current otherwise it is called practical current source.

Fig.3. Ideal and practical current source

Above figures show the characteristics of ideal and practical voltage and current source. For practical due to presence of internal resistance there will be some slight drop in response. In practical voltage source internal resistance is connected in series while for practical current source it is connected in parallel with current source.

**Independent and dependent source:**

Fig.4. Types of different sources

**Independent source: **These are those sources in which any electrical quantity does not depend upon any other electrical quantity like voltage, current.

Fig.5. Independent voltage source

**Dependent source: **It is defined as a source in which any electrical quantity depends upon other electrical quantity like voltage, current.

Fig.6. Independent current source

**5. Ohm’s law: **It is defined as in any closed circuit if the physical properties like pressure, temperature etc. are not changed then voltage drop in the circuit is directly proportional to the current flow in the circuit.

**Applications of ohm’s law:**

- Applicable to linear circuits only.
- Applicable to bilateral elements only.
- Use for determining current in the circuit.
- Determine voltage current relationship in the circuit.

__Summary__

We have discussed following topics as in chapter-1.

- Introduction to dc and ac circuits.
- Various types of elements.
- Differentiation between ac and dc supply.
- Different types of sources used in electric circuits.
- Dependent and independent sources.
- Ideal and practical voltage source.
- Ideal and practical current source.
- Introduction to ohm’s laws.
- Limitations of ohm’s law.

__Practice Questions__

**Q.1** Why we need electricity and what are the types of supply?

**Ans. **As we need electricity in our daily life we require to turn on our home appliances like television, drilling machine, refrigerator etc. so we require electricity. Without it there is no life. There are two types of supply such ac and dc.

**Q.**2 What are the types of elements?

**Ans.** There are different types of elements such as

- Active and passive
- Linear and non-linear
- Unilateral and bilateral

**Q.3** What is the difference between ideal and practical voltage source?

**Ans. **An ideal voltage source is that source in which any variation in current does not impact on voltage source otherwise it is called practical voltage source.

**Q.4** What do you mean by dependent source?

**Ans**. It is defined as a source in which any electrical quantity depends upon other electrical quantity like voltage, current.

## Lecture Topic 1.1.2

__Introduction to Kirchhoff’s Laws__

**Introduction: **Kirchhoff’s laws were introduced by Gustav Kirchhoff in 1845. They introduced Kirchhoff’s laws which were based on conservation of energy and conservation of charge. There are basically two types of laws such as Kirchhoff’s voltage law and Kirchhoff’s current law.

**Kirchhoff’s voltage law: **It states that in any closed loop circuit the algebraic sum of all voltage drops is zero.

**ΣV = 0**

**Or**

In other words, in any closed loop path the algebraic sum of all voltage drops and total emf sources is zero.

**ΣV + Σemf = 0**

__Introduction to Kirchhoff’s Laws__

**Introduction: **Kirchhoff’s laws were introduced by Gustav Kirchhoff in 1845. They introduced kirchhoff’s laws which were based on conservation of energy and conservation of charge. There are basically two types of laws such as kirchhoff’s voltage law and kirchhoff’s current law.

**Kirchhoff’s voltage law: **It states that in any closed loop circuit the algebraic sum of all voltage drops is zero.

**ΣV = 0**

**Or**

In other words, in any closed loop path the algebraic sum of all voltage drops and total emf sources is zero.

**ΣV + Σemf = 0**

From above it is shown that

**ΣV = Σ(V1 + V2 + V3)**

**Exp. **Find out the current in the circuit by using kvl.

Sol. We will consider that when we go from (-) polarity to (+) polarity we will take value as positive.

45-5xI-10xI-7.5xI = 0

(22.5×10^3)x I = 45

I = 45/( 22.5×10^3)

I = 2mA **Ans**

__Summary__

- Introduction to kirchhoff’s law.
- Studied about different terminologies used in kirchhoff’s law.
- Various examples based on kirchhoff’s voltage law.
- Difference between KVL & KCL.
- We have learnt applications of kirchhoff’s law.

__Practice Question__

**Q.1** What do you mean by kvl?

**Ans. **In any closed loop circuit the algebraic sum of potential drop is zero. It is based on conservation of energy.

**Q.2** What are the applications of kirchhoff’s law?

**Sol. **We can

- Evaluate current in any loop.
- Measure voltage across any branch
- Reduce complexity of the circuit.

## Lecture Topic [1.1.3]

__Introduction to Kirchhoff’s Laws__

**Kirchhoff’s current law:** It states that at any node the algebraic sum of currents is zero.

**ΣI = 0**

**Or**

At any node or junction the algebraic sum of incoming voltage and outgoing voltage is zero.

**ΣIinc = ΣIout**

Fig.1 KCL circuit

From figure we will consider incoming current is positive and outgoing current is negative.

**i1+i2+i6 = i3 +i4 +i5**

**Exp. **Find current I3 in given figure if I1 = 3A, I2 =5A, I4 = 1A

**Sol. **We will write equation:

I1 +I2 = I3 +I4

3 + 5 = I3 + 1

I3 = 7Amp **Ans**

__Summary__

- Introduction to Kirchhoff’s law.
- Studied about different terminologies used in Kirchhoff’s law.
- Various examples based on Kirchhoff’s current law.
- Difference between KVL & KCL.
- We have learnt applications of Kirchhoff’s law.

__Practice Question__

**Q.1** Explain kcl in your own statement.

**Ans.** It states that at any node the algebraic sum of currents is zero. It is based on conservation of charge.

**Q.2** What are the applications of Kirchhoff’s law?

**Sol. **We can

- Evaluate current in any loop.
- Measure voltage across any branch
- Reduce complexity of the circuit.

## Lecture Topic [1.1.4]

** Analysis of Series RL Circuit**:

A circuit that contains a pure resistance R ohms connected in series with a coil having a pure inductance of L (Henry) is known as** RL Series Circuit**. When an AC supply voltage V is applied, the current, I flows in the circuit.

So, IR and IL will be the current flowing in the resistor and inductor respectively, but the amount of current flowing through both the elements will be same as they are connected in series with each other. The circuit diagram of RL Series Circuit is shown below:

Fig.1 Series RL circuit

Where,

- VR – voltage across the resistor R
- VL – voltage across the inductor L
- V – Total voltage of the circuit

###### Phasor Diagram of the RL Series Circuit

The phasor diagram of the RL Series circuit is shown below:

Fig.2 Phasor dig of RL circuit

The following steps are given below which are followed to draw the phasor diagram step by step:

- Current I is taken as a reference.
- The Voltage drop across the resistance VR = IR is drawn in phase with the current I.
- The voltage drop across the inductive reactance VL =IXL is drawn ahead of the current I. As the current lags voltage by an angle of 90 degrees in the pure Inductive circuit.
- The vector sum of the two voltages drops VR and VL is equal to the applied voltage V.

Where,

Z is the total opposition offered to the flow of alternating current by an RL Series circuit and is called impedance of the circuit. It is measured in ohms (Ω).

###### Phase Angle

In RL Series circuit the current lags the voltage by 90 degrees angle known as phase angle. It is given by the equation:

__Series RC Circuit Analysis:__

A circuit that contains pure resistance R ohms connected in series with a pure capacitor of capacitance C farads is known as** RC Series Circuit.** A sinusoidal voltage is applied and current I flows through the resistance (R) and the capacitance (C) of the circuit.

Fig.3 Series RC circuit

The following steps are used to draw the phasor diagram of RC Series circuit

- Take the current I (r.m.s value) as a reference vector
- Voltage drop in resistance VR = IR is taken in phase with the current vector
- Voltage drop in capacitive reactance VC = IXC is drawn 90 degrees behind the current vector, as current leads voltage by 90 degrees (in the pure capacitive circuit)
- The vector sum of the two voltage drops is equal to the applied voltage V (r.m.s value).

###### Phase angle

From the phasor diagram shown above, it is clear that the current in the circuit leads the applied voltage by an angle ϕ and this angle is called the **phase angle**.

__Summary__

We have discussed following topics such as:

1. Analysis of RL Circuit(Phasor Diagram , Impedance, and Current in RL Circuit)

2. Analysis of RC Circuit (Phasor Diagram , Impedance,

and Current in RC Circuit)

## Lecture Topic [1.1.5]

__Series RLC Circuit Analysis__

When a pure resistance of R ohms, a pure inductance of L Henry and a pure capacitance of C farads are connected together in series combination with each other then **RLC Series Circuit** is formed. As all the three elements are connected in series so, the current flowing through each element of the circuit will be the same as the total current I flowing in the circuit.

Fig.1. Series RLC Circuit

In the RLC Series circuit

**XL = 2πfL** and** XC = 1/2πfC**

When the AC voltage is applied through the RLC Series circuit the resulting current I flows through the circuit, and thus the voltage across each element will be:

- VR = IR that is the voltage across the resistance R and is in phase with the current I.
- VL = IXL that is the voltage across the inductance L and it leads the current I by an angle of 90 degrees.
- VC = IXC that is the voltage across capacitor C and it lags the current I by an angle of 90 degrees.

**Phasor Diagram of RLC Series Circuit**

The phasor diagram of the RLC series circuit when the circuit is acting as an inductive circuit that means (VL>VC) is shown below and if (VL< VC) the circuit will behave as a capacitive circuit.

__Steps to draw phasor diagram:__

- Take current I as the reference as shown in the figure above
- The voltage across the inductor L that is VL is drawn leads the current I by a 90-degree angle.
- The voltage across the capacitor c that is Vc is drawn lagging the current I by a 90-degree angle because in capacitive load the current leads the voltage by an angle of 90 degrees.
- The two vector VL and VC are opposite to each other.

It is the total opposition offered to the flow of current by an RLC Circuit and is known as **Impedance** of the circuit.

**Phase Angle**

From the phasor diagram, the value of phase angle will be

**The three cases of RLC Series Circuit**

- When XL > XC, the phase angle ϕ is positive. The circuit behaves as RL series circuit in which the current lags behind the applied voltage and the power factor is lagging.
- When XL < XC, the phase angle ϕ is negative, and the circuit acts as a series RC circuit in which the current leads the voltage by 90 degrees.
- When XL = XC, the phase angle ϕ is zero, as a result, the circuit behaves like a purely resistive circuit. In this type of circuit, the current and voltage are in phase with each other. The value of the power factor is
**unity**.

###### Impedance Triangle of RLC Series Circuit

When the quantities of the phasor diagram are divided by the common factor I then the right angle triangle is obtained known as impedance triangle. The impedance triangle of the RL series circuit, when (XL > XC) is shown below:

If the inductive reactance is greater than the capacitive reactance than the circuit reactance is inductive giving a **lagging phase angle otherwise it will be called as leading phase angle.**

Fig.3. Impedance Triangle

###### Applications of RLC Series Circuit

The following are the application of the RLC circuit:

- It acts as a variable tuned circuit
- It acts as a low pass, high pass, bandpass, bandstop filters depending upon the type of frequency.
- The circuit also works as an oscillator
- Voltage multiplier and pulse discharge circuit

## Lecture Topic 1.2.1

__Introduction to Magnetic Circuits__

- A magnetic circuit is made up of one or more closed loop paths containing a magnetic flux. The flux is usually generated by permanent magnets or electromagnets and confined to the path by magnetic cores consisting of ferromagnetic materials like iron, although there may be air gaps or other materials in the path.
- Magnetic circuits are employed to efficiently channel magnetic fields in many devices such aselectricmotors,generators,transformers,relays,liftingelectromagnets,SQUIDs,galvanometers, and magnetic recording heads.
- The concept of a “magnetic circuit” exploits a one-to-one correspondence between the equations of the magnetic field in a non saturated ferromagnetic material to that of an electrical circuit.
- Using this concept the magnetic fields of complex devices such as transformers can be quickly solved using the methods and techniques developed for electrical circuits.

Some examples of magnetic circuits are:

· horseshoe magnet with iron keeper (low-reluctance circuit)

· horseshoe magnet with no keeper (high-reluctance circuit)

· electric motor (variable-reluctance circuit)

· some types of pickup cartridge (variable-reluctance circuits)

A **magnetic circuit** is made up of magnetic materials having high permeability such as iron, soft steel, etc. **Magnetic circuits** are used in various devices like electric motor, transformers, relays, generators galvanometer, etc.

Fig.1 Magnetic Circuit

Consider a solenoid having N turns wound on an iron core. The magnetic flux of ø Weber sets up in the core when the current of I ampere is passed through a solenoid as shown above.

Let, l = mean length of the magnetic circuit

A = cross-sectional area of the core

µr = relative permeability of the core

Now the flux density in the core material

Magnetising force in the core According to work law, the work done in moving a unit pole once round the magnetic circuit is equal to the ampere-turns enclosed by the magnetic circuit.

According to work law, the work done in moving a unit pole once round the magnetic circuit is equal to the ampere-turns enclosed by the magnetic circuit.

The above equation explains the following points:

- Directly proportional to the number of turns (N) and current (I).It shows that the flux increase if the number of turns or current increases and decreases when either of the two quantitydecreases.NI is the magnetomotive force (MMF).
- Inversely proportional tol/aµ0µr,where(l/aµ0µr)is known as reluctance. The lower the reluctance, the higher will be the flux and vice- verse.

__Different terms used in Magnetic Circuits__

**Magnetic Field or Magnetic Induction (B)**

Magnet or Electromagnet produces a Magnetic field. The field where the magnet attracts or repels magnetic materials such as iron, steel, etc. it may be defined as a force on a moving charge,

**F = q x v x B**

Where

· F = Force,

· V = Speed of Particles,

· B = magnitude of the field.

__Magnetic Field Strength (H)__

The amount of magnetizing force (how much force it has to magnetize, magnetic materials such as iron, steel, etc) is called Magnetic field strength which is denoted by (H). It is inversely proportional to the length of wire and directly proportional to the current passing through it. The SI unit of Magnetic Field Strength is Ampere/meter (A/m) and it is a vector quantity and the SI formula for Magnetic Field strength is

**H = NI / 1c**

Where 1c = magnetic path in meter.

__Magnetic Flux (Φ)__

In simple words, Magnetic field x area perpendicular to the magnetic field (B) is called Magnetic Flux which is denoted by Φ or Φm or ΦB. Or it is the amount of magnetic field or magnetic lines of force passing through a surface like conducting area, space, air, etc. The SI Unit of magnetic flux is Wb (Weber). The Formula for finding magnetic flux in the SI system is;

**Φ = BAc**

Where

Ac = area in m2

And CGS unit and formula for Magnetic Flux is Maxwell (M) and Φ = BAc Ac = area in cm2 respectively.

__Magnetization (M)__

The state of a material being magnetized or the process in which magnetic materials are magnetized. It is the density of permanent magnet or electromagnet dipole moments in magnetic materials. Or the magnetic moment (m) per unit volume (v) by a magnetic field is called Magnetization. The SI Unit of Magnetization is Ampere/meter (A/m) and it is also a vector quantity. The SI formula for Magnetization is

**M = m/V**

Where,

m = Total magnetic moment

And V= volume in m3.

The CGS unit and formula of Magnetization is Emu/cm3 and M = m/V respectively, where, m = Total magnetic moment, V = volume in cm3 and EMU = Electromagnetic units. It may also be defined in term of M = (N/V) x m → M = nm ……. (N/V) = n. Where, “m” is the magnetic moment and “n” is the number density of magnetic moments.

__Magnetic Permeability of vacuum__

§ It is the amount of resistance encountered to the magnetic field when forming in a vacuum.

§ The SI unit of Permeability is (H·m−1), or Newton per ampere squared (N·A−2). The SI unit and formulas of Magnetic Permeability of vacuum is Newton/Ampere2 and **µ○ = 4πx10-7 ≈ 1.2566370614 H·m−1** respectively. The CGS unit of magnetic permeability of vacuum is 1.

__Inductance (L)__

Inductance is the property of conductor, coil or wire which opposes the change of current flowing through it. The change of current flowing through a conductor produces a voltage called Back EMF or Electro motive force.

Even The change of current flowing through a conductor or coil produces voltage through it which is called Self induced EMF and in any nearby coils or conductors which is called Mutual inductance. The SI unit of Inductance (L) is Henry “H” and formula is

**L = µ○ µ N2 Ac/1c**

Where

· N = Turns

· Ac = Area in m2

· 1c = magnetic path in meter

CGS unit and formula of Inductance is Henry “H” (Joseph Henry) and **L = 0.4π µN2Ac/1c x10-8 **respectively

where;

· L = Inductance

· N = Turns

· Ac = Area in cm2

· 1c = magnetic path in cm.

**Self Inductance formula**

**L = µ○ (N2xA)/l**

Where:

· L = in Henries

· μο = the Permeability of Free Space (4.π.10-7)

· N = the Number of turns

· A = the Inner Core Area (π.r 2) in m2

· l = the length of the Coil in meters

**Mutual Inductance formula**

**M = μο μrN1N2A/l**

Where:

· µo = the permeability of free space (4.π.10-7)

· µr = the relative permeability of the soft iron core

· N = in the number of coil turns

· A = in the cross-sectional area in m2

· l = the coils length in meters

__Voltage or E.M.F (V)__

The Electric Potential Difference between two points is called Voltage. Or the work done per unit charge in a static electric field to move the charge between two points, so the equation becomes as

Where;

· V = Voltage

· E = Energy in joules

· q = Charge in Coulombs

Or the electric potential energy per unit charge is called Voltage.

In Ohm’s Law, Voltage = V = I x R, Where I = Current in amperes and R = Resistance in Ohms (Ω)

The SI unit of Voltage is the Volt (V) or Joules per Coulomb. Where 1V = 1Joule/1Coulomb

The SI formula of Voltage is

**V = -N dΦ/dt**

Where;

· N = number of coil Turns

· dΦ = rate of the Change in flux

· t = time

__Laws of Electromagnetic Induction__

**Faraday’s law of electromagnetic induction** (referred to as **Faraday’s law**) is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF). This phenomenon is known as electromagnetic induction.

Faraday’s law states that a current will be induced in a conductor which is exposed to a changing magnetic field. Lenz’s law of electromagnetic induction states that the direction of this induced current will be such that the magnetic field created by the induced current ** opposes** the initial changing magnetic field which produced it. The direction of this current flow can be determined using Fleming’s right-hand rule.

__Faraday’s First Law__

Any change in the magnetic field of a coil of wire will cause an emf to be induced in the coil. This emf induced is called induced emf and if the conductor circuit is closed, the current will also circulate through the circuit and this current is called induced current.

Method to change the magnetic field:

1. By moving a magnet towards or away from the coil

2. By moving the coil into or out of the magnetic field

3. By changing the area of a coil placed in the magnetic field

4. By rotating the coil relative to the magnet

Faraday’s Second Law

It states that the magnitude of emf induced in the coil is equal to the rate of change of flux that linkages with the coil. The flux linkage of the coil is the product of the number of turns in the coil and flux associated with the coil.

__Applications of Faraday’s Law__

- Power transformers function based on Faraday’s law
- The basic working principle of the electrical generator is Faraday’s law of mutual induction.
- The Induction cooker is the fastest way of cooking. It also works on the principle of mutual induction. When current flows through the coil of copper wire placed below a cooking container, it produces a changing magnetic field. This alternating or changing magnetic field induces an emf and hence the current in the conductive container, and we know that the flow of current always produces heat in it.
- Electromagnetic Flow Meter is used to measure the velocity of certain fluids. When a magnetic field is applied to an electrically insulated pipe in which conducting fluids are flowing, then according to Faraday’s law, an electromotive force is induced in it. This induced emf is proportional to the velocity of fluid flowing.
- Form bases of Electromagnetic theory, Faraday’s idea of lines of force is used in well known Maxwell’s equations. According to Faraday’s law, change in magnetic field gives rise to change in electric field and the converse of this is used in Maxwell’s equations.

__Applications__

Magnetic circuits are employed to efficiently channel magnetic fields in many devices such as

- electric motors
- generators
- transformers and relays
- lifting electromagnets
- SQUIDs
- galvanometers and magnetic recording heads.

__Summary__

In this section we have focused on following topics such as:

- Introduction to magnetic circuit.
- Different terms used in magnetic circuit.
- Concept of mmf and magnetic flux.
- Concept of magnetic permeance and magnetic permeability.
- Types of magnetic circuit.
- Applications of magnetic circuit.

## Lecture Topic 1.2.2

__Self inductance and Mutual inductance__

**Self-inductance** or in other words inductance of the coil is defined as the property of the coil due to which it opposes the change of current flowing through it. Inductance is attained by a coil due to the self-induced emf produced in the coil itself by changing the current flowing through it.

If the current in the coil is increasing, the self-induced emf produced in the coil will oppose the rise of current, that means the direction of the induced emf is opposite to the applied voltage.__Self inductance and Mutual inductance__

**Self-inductance** or in other words inductance of the coil is defined as the property of the coil due to which it opposes the change of current flowing through it. Inductance is attained by a coil due to the self-induced emf produced in the coil itself by changing the current flowing through it.

If the current in the coil is increasing, the self-induced emf produced in the coil will oppose the rise of current, that means the direction of the induced emf is opposite to the applied voltage.

If the current in the coil is decreasing, the emf induced in the coil is in such a direction as to oppose the fall of current; this means that the direction of the self-induced emf is same as that of the applied voltage. Self-inductance does not prevent the change of current, but it delays the change of current flowing through it.

This property of the coil only opposes the changing current (alternating current) and does not affect the steady current that is (direct current) when flows through it. The unit of inductance is **Henry** (H).

**Expression For Self Inductance**

You can determine the self-inductance of a coil by the following expression.

The above expression is used when the magnitude of self-induced emf (e) in the coil and the rate of change of current (dI/dt) is known.

Putting the following values in the above equations as e = 1 V, and dI/dt = 1 A/s then the value of Inductance will be L = 1 H.

Hence, from the above derivation, a statement can be given that a coil is said to have an inductance of 1 Henry if an emf of 1 volt is induced in it when the current flowing through it changes at the rate of 1 Ampere/second.

The expression for Self Inductance can also be given as:

From the above discussion, the following points can be drawn about Self Inductance

· The value of the inductance will be high if the magnetic flux is stronger for the given value of current.

· The value of the Inductance also depends upon the material of the core and the number of turns in the coil or solenoid.

· The higher will be the value of the inductance in Henry, the rate of change of current will be lower.

· 1 Henry is also equal to 1 Weber/ampere

The solenoid has large self-inductance.

** Mutual Inductance** :

**It** is defined as the property of the coil due to which it opposes the change of current in the other coil, or you can say in the neighbouring coil.

When the current in the neighbouring coil changes, the flux sets up in the coil and because of this, changing flux emf is induced in the coil called Mutually Induced emf and the phenomenon is known as **Mutual Inductance**.

Let us understand the phenomenon of Mutual Inductance by considering an example as shown in the above figure.

Two coils namely coil A and coil B are placed nearer to each other. When the switch S is closed, and the current flows in the coil, it sets up the flux φ in the coil A and emf is induced in the coil and if the value of the current is changed by varying the value of the resistance (R), the flux linking with the coil B also changes because of this changing current.

Fig.2. Mutual Inductance

Thus this phenomenon of the linking flux of the coil A with the other coil, B is called **Mutual Inductance**.

For determining the Mutual Inductance between the two coils, the following expression is used

This expression is used when the magnitude of mutually induced emf in the coil and the rate of change of current in the neighbouring coil is known.

If emf = 1 volt and dI1/dt = 1 ampere then putting this value in the equation (1) we get the value of mutual inductance as M=1 Henry

Hence, from the above statement, you can define Mutual Inductance as “the two coils are said to have a mutual inductance of one Henry if an emf of 1 volt is induced in one coil or say primary coil when the current flowing through the other neighbouring coil or secondary coil is changing at the rate of 1 ampere/second”.

The value of Mutual Inductance (M) depends upon the following factors

· Number of turns in the secondary or neighboring coil

· Cross-sectional area

· Closeness of the two coils

**Mutual Coupling In the Magnetic Circuit**

When on a magnetic core, two or more than two coils are wound, the coils are said to be mutually coupled. The current, when passed in any of the coils wound around the magnetic core, produces flux which links all the coils together and also the one in which current is passed. Hence, there will be both self-induced emf and mutual induced emf in each of the coils.

The best example of the mutual inductance is the transformer, which works on the principle of **Faraday’s Law of**

**Electromagnetic Induction**.

Faraday’s law of electromagnetic induction states that “the magnitude of voltage is directly proportional to the rate of change of flux.” which is explained in the topic Faraday’s Law of Electromagnetic Induction.

__Summary__

In this section we have focused on following topics such as:

- Introduction to faraday’s law of electromagnetic induction.
- Application of faraday’s law.
- Concept of Lenz law.
- Self inductance phenomenon.
- Mutual inductance phenomenon.
- Coefficient of coupling.

__Numerical Practice__

**1.**** ****A long solenoid has 500 turns. When a current of 2 A is passed through it, the resulting magnetic flux linked with each turn of the solenoid is 4×10−3 Wb. Find the self-inductance of the solenoid.**

**2.**** ****Two coils connected in series–aiding fashion have a total inductance of 600mH. When connected in a series-opposing configuration, the coils have a total inductance of 400mH. If the inductance of one coil is three times the other , Find L1,L2 ,M and the value of coupling coefficient.**

__Practice Questions__

Q.1** **Explain the farday’s law of electromagnetic induction .

Q.2 Explain the difference between self inductance and mutual inductance.

## Lecture Topic 1.2.3

__Transformer__

A transformer is defined as a passive electrical device that transfers electrical energy from one circuit to another through the process of electromagnetic induction. It is most commonly used to increase (‘step up’) or decrease (‘step down’) voltage levels between circuits.

**Working principle of electrical power transformer** is very easy, it is similar to that of mutual induction. A **transformer** is a static (or stationary) piece of apparatus by means of which electric power in one circuit is transformed into electric power of the same frequency in another circuit.

- It can raise or lower the voltage in a circuit but with a corresponding decrease or increase in current.
- The physical basis of a power transformer is mutual induction between two circuits linked by a common magnetic flux. In its simplest form, it consists of two inductive coils which are electrically separated but magnetically linked through a path of low reluctance as shown in the figure below.
- The two coils possess high mutual inductance. If one coil is connected to a source of alternating voltage, an alternating flux is set up in the laminated core, most of which is linked with the other coil in which it produces mutually-induced e.m.f. (according to Faraday’s Laws of Electromagnetic Induction = M.dI/dt).
- If the second coil circuit is closed, a current flow in it and so electric energy is transferred (entirely magnetically)from the first coil to the second coil.
- The first coil, in which electric energy is fed from the a.c.supply mains is called primary winding and the other from which energy is drawn out, is called secondary winding.

Fig.1 Transformer

- The power transformer consists of two coils having mutual inductance and a laminated steel core. The two coils are insulated from each other and the steel core. Other necessary parts are some suitable container for assembled core and windings; a suitable medium for insulating the core and its windings from its container; suitable bushings(either of porcelain, oil-filled or capacitor-type) for insulating and bringing out the terminals of windings from the tank.
- In all types of power transformers, the core is constructed of transformer sheet steel laminations assembled to provide a continuous magnetic path with a minimum of air-gap included.
- The steel used is of high silicon content, sometimes heat treated to produce a high permeability and a low hysteresis loss at the usual operating flux densities. The eddy current loss is minimised by laminating the core, the laminations being insulated from each other by a light coat of core-plate varnish or by an oxide layer on the surface.
- The thickness of laminations varies from 0.35 mm for a frequency of 50 Hz to 0.5 mm for a frequency of 25 Hz. The core laminations (in the form of strips) are joined as shown in Figure.

It is seen that the joints in the alternate layers are staggered in order to avoid the presence of narrow gaps right through the cross-section of the core. Such staggered joints are said to be ‘imbricate.

The two types are known as

(i) core-type transformer (ii) shell-type transformers

**Core Types Transformer:** Another recent development is spiral-core or wound-core type, the trade name being Spiral Core transformer. In the so called **core type transformers**, the windings surround a considerable part of the core whereas, in shell type transformers, the core surrounds a considerable portion of the windings as shown schematically in the figures(a) and (b) respectively.

In the simplified diagram for the core type transformers, the primary and secondary winding are shown located on the opposite legs (or limbs) of the core, but in actual construction, these are always interleaved to reduce leakage flux.

As shown in Figure, half the primary and half the secondary winding have been placed side by side or concentrically on each limb, not primary on one limb (or leg) and the secondary on the other.

**Shell Types Transformer:**

HV and LV windings are wound longitudinally along the core alternately. The HV coils are sandwiched between two LV coils as shown in the figure below

ince, both HV and LV coils are wound on the central limb surface, the quantity of conductor required for woundings of shell type transformer is less than that of a similar core type transformer.

Fig.3 Shell type Transformer

The design of shell-type is a bit complex compared to core-type due to its winding structure. Any defect in the inner windings can only be attended after removing all outer windings hence the winding maintenance jobs are very hard in shell type transformer.

**Transformation Ratio**

The **transformer transformation ratio** or **transformer turns ratio** (K) is the quotient value obtained by dividing the number of **turns** of the primary winding (N1) and the number of **turns** of the secondary winding (N2).

Then **K = N1/N2**

**K = V1/V2 = I2/I1 =N1/N2**

**Voltage Transformation Ratio**

A transformer with an equal number of turns on its primary and secondary windings will have a secondary voltage only slightly less than the primary applied voltage, and its voltage ratio is said to be 1:1. If, however, the secondary winding have only one-half as many turns as the primary winding, the secondary voltage will be only one-half as great as the primary voltage. The voltage ratio then will be 2:1. The primary and secondary generated electromotive forces are proportional to the primary and secondary turns respectively.

__Summary__

**As we have discussed following topics:**

- Faraday’s Law of Electromagnetic Induction
- Types of induced emf
- Self induced emf
- Mutual induced emf
- Introduction of single phase transformer
- Working Principle of Single Phase Transformer
- Different types of transformer
- Step up transformer
- Step down transformer
- Construction of Transformer
- core type
- shell type
- Transformation ratio

__FREQUENTLY ASKED QUESTIONS__

** 1.) What does a single-phase mean?**

Answer:- A single-phase system or circuit which generates or uses single alternating voltage

**2.) Do houses use single-phase supply?**

Answer:- Generally, homes are supplied with single-phase supply

**3.) On which principles does the single-phase transformer operate?**

Answer:- Faraday’s law of Electromagnetic Induction and Mutual Induction

**4.) What is the transformer “Turns Ratio”?**

Answer:- NP/NS = VP/VS = n = Turns Ratio

**5.) Give two uses of a single-phase transformer**

Answer:- In television sets for voltage regulation

To step-up power in home inverters

## Lecture Topic 1.2.4

__Losses in transformer__

There are various losses exist in a transformer which occur due to the heat present in the losses. These losses reduce the efficiency of the working of transformer.

Below chart shows the description of transformer:

**Core losses or Iron losses**

• Iron losses are caused by the alternating flux in the core of the transformer as this loss occurs in the core it is also known as **Core loss**. Iron loss is further divided into hysteresis and eddy current loss.

• Eddy current loss and hysteresis loss depend upon the magnetic properties of the material used for the construction of core.

• Hence these losses are also known as **core losses** or **iron losses**.

**Hysteresis loss in transformer**: Hysteresis loss is due to reversal of magnetization in the transformer core. This loss depends upon the volume and grade of the iron, frequency of magnetic reversals and value of flux density.

Where

· KȠ is a proportionality constant which depends upon the volume and quality of the material of the core used in the transformer,

· f is the supply frequency,

· Bmax is the maximum or peak value of the flux density.

The iron or core losses can be minimized by using silicon steel material for the construction of the core of the transformer.

**Eddy Current Loss**

When the flux links with a closed circuit, an emf is induced in the circuit and the current flows, the value of the current depends upon the amount of emf around the circuit and the resistance of the circuit.

Since the core is made of conducting material, these EMFs circulate currents within the body of the material.

These circulating currents are called **Eddy Currents**. They will occur when the conductor experiences a changing magnetic field.

As these currents are not responsible for doing any useful work, and it produces a loss (I2R loss) in the magnetic material known as an **Eddy Current Loss**.

The eddy current loss is minimized by making the core with thin laminations.

The equation of the eddy current loss is given as:

Where,

· Ke – coefficient of eddy current. Its value depends upon the nature of magnetic material like volume and resistivity of core material, the thickness of laminations

· Bm – maximum value of flux density in wb/m2

· T – thickness of lamination in meters

· F – frequency of reversal of the magnetic field in Hz

· V – the volume of magnetic material in m3

Copper Loss Or Ohmic Loss

These losses occur due to ohmic resistance of the transformer windings. If I1 and I2 are the primary and the secondary current. R1 and R2 are the resistance of primary and secondary winding then the copper losses occurring in the primary and secondary winding will be I12R1 and I22R2 respectively.

Therefore, the total copper losses will be

These losses varied according to the load and known hence it is also known as variable losses. Copper losses vary as the square of the load current.

Stray Loss

The occurrence of these stray losses is due to the presence of leakage field.

The percentage of these losses are very small as compared to the iron and copper losses so they can be neglected.

Dielectric Loss

Dielectric loss occurs in the insulating material of the transformer that is in the oil of the transformer, or in the solid insulations.

When the oil gets deteriorated or the solid insulation gets damaged, or its quality decreases, and because of this, the efficiency of the transformer gets affected.

__Summary__

**As we have discussed following topics:**

- Construction of transformer
- Ideal transformer
- Losses in transformer
- Core losses or Iron losses
- Copper loss
- Stray losses
- Dielectric loss
- How to reduced core losses
- How to reduced hysteresis losses

## Lecture Topic 1.2.5

__Transformer Efficiency:__

The **Efficiency** of the transformer is defined as the ratio of useful output power to the input power. The input and output power are measured in the same unit. Its unit is either in Watts (W) or KW. **Transformer efficiency** is denoted by **Ƞ**.

Where,

· V2 – Secondary terminal voltage

· I2 – Full load secondary current

· Cosϕ2 – power factor of the load

· Pi – Iron losses = hysteresis losses + eddy current losses

· Pc – Full load copper losses = I22Res

Consider, the x is the fraction of the full load. The efficiency of the transformer regarding x is expressed as

given power factor the Transformer efficiency depends upon the load current I2. In equation (1), the numerator is constant and the transformer efficiency will be maximum if the denominator with respect to the variable I2 is equated to zero.

i.e Copper losses = Iron losses

Thus, the transformer will give the maximum efficiency when their copper loss is equal to the iron loss.

From equation (2) the value of output current I2 at which the transformer efficiency will be maximum is given as

If x is the fraction of full load KVA at which the efficiency of the transformer is maximum then,

Copper losses = x2Pc (where Pc is the full load copper losses)

Iron losses = Pi

**For maximum efficiency**

**x2 Pc = Pi**

Therefore

Thus, output KVA corresponding to maximum efficiency

Putting the value of x from the above equation (3) in equation (4) we will get,

The above equation (5) is the maximum efficiency condition of the transformer.

**Transformer Voltage Regulation**

We have seen in this series of tutorials about the transformer, that when the primary winding of a transformer is energized, it produces a secondary voltage and current at an amount determined by the transformers turns ratio, (TR). So if a single-phase transformer has a step down turns ratio of 2:1 and 240V is applied to the high voltage primary winding, we would expect to see an output terminal voltage on the secondary winding of 120 VAC because we have assumed it to be an ideal transformer.

However in the real world this is not always true as being a wound magnetic circuit, all transformers suffer from losses consisting of I2R copper losses and magnetic core losses which would reduce this ideal secondary value by a few percent to say 117 VAC, and this is normal. But there is also another value related to transformers (and electrical machines) which also has an affect on this secondary voltage value when the transformer is supplying full power, and this is called “regulation”.

**Voltage Regulation** of single-phase transformers is the percentage (or per unit value) change in its secondary terminal voltage compared to its original no-load voltage under varying secondary load conditions. In other words, regulation determines the variation in secondary terminal voltage which occurs inside the transformer as a result of variations in the transformers connected load thereby affecting its performance and efficiency if these losses are high and the secondary voltage becomes too low.

When there is no-load connected to the transformers secondary winding, that is its output terminals are open-circuited, there is no closed-loop condition, so there is no output load current (IL = 0) and the transformer acts as one single winding of high self-inductance. Note that the no-load secondary voltage is a result of the fixed primary voltage and the turns ratio of the transformer.

Loading the secondary winding with a simple load impedance causes a secondary current to flow, at any power factor, through the internal winding of the transformer. Thus voltage drops due to the windings internal resistance and its leakage reactance causes the output terminal voltage to change.

A transformers voltage regulation change between its secondary terminal voltage from a no-load condition when IL = 0, (open circuit) to a fully-loaded condition when IL = IMAX (maximum current) for a constant primary voltage is given as:

__Numerical Practice__

**1.**** **** **A single-phase transformer has 400 primary and 1000 secondary turns. The net cross-sectional area of the core is 60 cm2. If the primary winding be connected to a 50-Hz supply at 520 V, calculate **(i) **the peak value of flux density in the core **(ii) **the voltage induced in the secondary winding.

**2.**** **A single phase transformer has 500 turns in the primary and 1200 turns in the secondary. The cross-sectional area of the core is 80 sq. cm. If the primary winding is connected to a 50 Hz supply at 500 V, calculate **(i) **Peak flux-density, and **(ii) **Voltage induced in the secondary.

__Summary__

**As we have discussed following topics:**

- Various types of losses
- Why is a transformer rated in KVA but not in KW?
- Transformer Efficiency
- Condition for maximum efficiency of transformer
- All Day Efficiency of a Transformer
- Voltage Regulation
- Expression of Voltage Regulation of Transformer
- Applications of transformer