EMF Equation OF Transformer | Turn voltage ratio of Transformer

EMF Equation Of Transformer

⇒ Let us consider an ideal transformer on no load, when an alternating voltage V1 is applied to the primary side of the transformer thus magnetizing current start flowing in the primary winding and flux Φ is generated in the transformer core.

⇒ The flux Φ is uniformly distributed around the transformer core and it is linked with both primary and secondary winding of the transformer.

Emf equation of the transformer

⇒ The flux Φ is alternating in nature therefore according to Faraday’s law of electromagnetic induction, emf is induced in the primary side or winding of the transformer.
PRIMARY INDUCED EMFWhere e1 is primary induced emf

⇒ According to Lenz’s law, the induced emf e1 is equal and opposite the supply voltage V1

⇒ Since the source given to the primary is sinusoidal therefore the flux produced is also sinusoidal in nature. Hence the sinusoidal flux Φ produced by the primary can be represented as.
Φ = Φm sin ωt

⇒ Putting the value of Φ we get
Emf equation of the transformerr

⇒ On differentiating the above equation we get

e1 = -ω N1 Φm cosωt

e= -2πf N1 Φm cosωt ……(since ω = 2πf )

e1 = 2πf N1 Φsin(ωt – 90.) . . . . . (1)

⇒ From equation 1 it is clear that the induced emf lags flux by 90 degree hence maximum value of induced emf will be
emax = 2πf N1 Φm

⇒ The rms value can be obtained by dividing the maximum value of induced emf by √2 therefore
Emf equation of the transformer

Voltage Transformation ratio

Note: From equation 1 it is clear that induced emf e1 in the primary winding lags behind the flux by 90o. Similarly, the induced emf e2 in the secondary lags behind the flux by 90o

Voltage Transformation Ratio (K)


The constant K is called as Voltage transformation ratio and it is given as:

For an ideal transformer, there is no voltage drop in the windings, therefore, E1 = E2 and V1 = V2.
Ideal transformer voltage ratio

If N2 > N1 and K >1 then the transformer is called as step up transformer.
If N1 > N2 and K< 2 then the transformer is called as step down transformer.

There are no losses in an ideal transformer therefore volt-ampere input of the primary will be always equal to the volt ampere of the secondary.
Ideal transformer volt ampere ratio

From above equation, it is clear that current is inversely proportional to the voltage i.e if we will increase the value of voltage than the value of current will decrease.

Synchronous Motor Working Principle

Synchronous Motor Working Principle

  • Electric Motor is an electromechanical device which transforms electric energy into mechanical energy.
  • According to their type of connection, electric motors are generally classified into the two types i.e single phase motor and three phase motor.
  • A synchronous motor is a 3 phase motor and it closely resembles 3 phase alternator.
  • 3 phase synchronous motor and 3 phase induction motor are most widely used  AC motor.
  • A synchronous motor is also called as doubly excited motor.

The synchronous motor consist of the two parts:

Stator: Stator is the armature winding. It consists of three phase star or delta connected winding and excited by 3 phase A.C supply.

Rotor: Rotor is a field winding. The field winding is excited by the separate D.C supply through the slip ring.
The construction of Rotor can be salient pole (projected pole) and non-salient pole (cylindrical pole) type.

3 phase Synchronous motor


Principle Of  Working Of Synchronous Motor

  • Synchronous motor work on the principle of magnetic locking.
  • When two unlike strong unlike magnets poles are brought together, there exists a tremendous force of extraction between those two poles. In such condition, the two magnets are said to be magnetically locked.



  • If now one of the two magnets is rotated, the other magnets also rotate in the same direction with the same speed due to the strong force of attraction.
  • This phenomenon is called as magnetic locking 

For magnetic locking condition, there must be two unlike poles and magnetic axes of this two poles must be brought very nearer to each other.


  • Consider a synchronous motor whose stator is wound for 2 poles.
  • The stator winding is excited with 3 phase A.C supply and rotor winding with D.C supply respectively. Thus two magnetic fields are produced in the synchronous motor.
  • When the 3 phase winding is supplied by 3 phase A.C supply than the rotating magnetic field or flux is produced.
  • This magnetic field or flux rotates in a space at a speed called synchronous speed.
  • The rotating magnetic field or rotating flux has fixed relationship between, the number of poles, the frequency of a.c supply and the speed of rotation.
  • The rotating magnetic field creates an effect which is similar to the physical rotation of magnets in space with a synchronous speed.
  • So for rotating magnetic field
    speed of Synchronous motor

    Where f = supply frequency
    P = Number of poles


Synchronous Motor Action

Synchronous Motor Action

  • Suppose the stator poles are N1 and S1 which are rotating at a speed of N and the direction of rotation be clockwise.
  • When the field winding on a rotor is excited by the D.C source, it produces the two stationary poles i.e N2 and S2.
  • To establish the magnetic locking between the stator and rotor poles the, unlike poles N1 and S2 or N2 and S1 should be brought near to each other.
  • As stator poles are rotating and due to magnetic locking the rotor poles will rotate in the same direction of rotating magnetic field as that of stator poles with the same speed Ns.
  • Hence synchronous motor rotates at only one speed that is synchronous speed.
  • The synchronous speed depends on the frequency therefore for constant supply frequency synchronous motor speed will be constant irrespective of the load changed.

Features of Synchronous Motor

  • It runs either at synchronous speed or not at all. That is while running it maintains a constant speed. The speed is independent of load.
  • It is not inherently self-starting. It has to be run at synchronous speed by some means before it can be synchronized to supply.
  • It can be operated under the wide range of power factors both lagging and leading.
  • It will stall if, while running, counter torque is increased beyond the maximum torque that machine can develop.
  • The speed of the synchronous motor can be controlled by inverter units.

Application of Synchronous Motor.

Synchronous motor finds various application for following services:

  • Power Factor correction
  • Voltage Regulation
  • Constant speed, Constant load drives

Power Factor Correction

  • Over excited synchronous motor having leading power factor are widely used for improving power factor for those power system which employs a large number of an induction motor.
  • Under excited synchronous motor having lagging power factor found application in fluorescent light welding etc.

Voltage Regulation

  • The voltage at the long transmission lines varies greatly when the large inductive load is present.
  • When line voltage decreases due to inductive load, motor excitation is increased thereby raising its power factor which compensates for the line drop.
  • If line voltage rises due to line capacitive effect, motor excitation is decreased thereby making its power factor lagging which helps to maintain the line voltage to its normal value.

Constant Speed Application

  • Because of their high efficiency and high-speed synchronous motor are well suited for loads where constant speed is required such as a centrifugal pump, blowers, line shaft, paper mills etc.


Speed Regulation Of DC Motor

Suppose Mohan and Ram go to school by cycle. One day Ram cycle get a puncture so he asked Mohan for help. Now Mohan has to carry the extra weight of Ram. So obviously the extra weight will reduce the speed of Mohan’s cycle. Therefore he has to exert some extra power to reach in time. This extra power in an electrical machine is known as torque.

So does the only factor of extra power matter to reach school in time…. the answer is no. Now you all are wondering why? Let me explain

The another factor that plays a vital role is the time taken by Mohan to maintain his usual speed after the increase in load. If he takes more time to adjust his speed he can’t reach school in time. That means less the variation in speed more chances to reach school on time.

  • The same theory works with DC motor i.e when the load is applied to the dc motor its speed decrease, but it is not desirable since in many application such as conveyors, lathe machine etc. we need constant speed motor.
  • So it is desirable that the difference between the no load to full load speed should be less.

Speed Regulation of DC motor

  • When we say DC motor is a self-regulating machine. This self-regulating effect is called as speed regulation. That means the motor will adjust its speed with the variation in the load.
  • The speed regulation is defined as the ratio of change in the speed from no load to full load to the speed corresponding to full load.
  • Numerically it is  expressed as
    Speed regulation of DC motor
  • Similarly, percentage speed regulation is defined as
    percentage speed regulation of dc motor

Note ⇒ The lower the percentage of regulation the more constant the speed of dc motor.

  • The EMF equation of DC motor is given by
    EMF equation of dc motor

    From the above equation, it is clear that back emf of DC motor is directly proportional to the speed of the DC motor.
  • If the load is added to the motor then the motor must produce more torque to overcome the added load and  T α Ia hence armature current also increase with the increase in the load.
  • To produce more torque the magnetic field of the pole must increase, and the increase in the field strength can be achieved when the armature speed decreases causing less back emf to be produced in the armature.
  • The decrease of the back emf allows more current to flow through the armature causing an increase in magnetic field strength.
  • In DC motor the speed regulation is proportional to the resistance of the armature.
  • The lower the armature resistance the better will be the speed regulation of the dc motor.

Speed regulation of various motors

DC shunt motor

  • The speed regulation of DC shunt motor is between 10 -15 %.
shunt motor speed characteristics

shunt motor speed characteristics

DC Series motor

  • The speed regulation of dc series motor is most inferior among all the dc motor.
  • The percentage of speed regulation is more than 35 %.

DC Cumulative compound motor

  • The speed regulation of DC cumulative compound motor is superior to that of dc series motor and inferior to DC shunt motor.
  • The percentage of speed regulation of Dc cumulative compound motor is between 25% – 30%.

DC differential compound motor

  • The speed regulation of a DC differential compound motor is superior among all the other motor.
  • The percentage of speed regulation of DC differential compound motor is between 3% -5%.

speed load characteristic of dc motors


Working Principle of D.C Motor

Working Principle of DC motor

  • The DC motor work on the principle when the current carrying conductor is placed in a magnetic field it experiences a mechanical force.
  • The force is experienced from high flux density to low flux density.
  • Magnitude of force experienced by the armature conductor is

    B = magnetic flux density produced by field winding
    L = Active length of the conductor
    I = Magnitude of the current carrying by the conductor
  • The direction of the force is determined by Fleming’s left-hand rule.


Fleming’s Left-hand Rule

  • When a current carrying conductor such as a wire attached to a circuit moves placed in a magnetic field, an electric current is induced in the wire due to Faraday’s law of induction.

Fleming's left-hand rule

  • The left hand is held with the thumb, first finger and second finger mutually perpendicular to each other than
  • The thumb is pointed in the direction of the motion of the conductor relative to the magnetic field i.e direction of the force.
  • The forefinger is pointed in the direction of the magnetic field.
  • The middle finger represents the direction of the induced or generated current within the conductor.


  • On the basis of their construction, there is no difference between a D.C generator and a D.C motor.
  • The same D.C machine can be used as the D.C motor and generator
  • As in D.C generator flux is created by field winding.
  • Armature winding act as a current carrying conductor.

A DC  motor is a machine which converts DC power or current into the mechanical output.



Here the input voltage is in the form of voltage and electrical and the mechanical output is in the form of torque (T) and speed  (ω).

In this section, we will study about the Principle of DC motor, its working, and rules related to DC machine.

Back EMF in D.C Motor

  • When the motor starts rotating its conductor will cut the magnetic flux produced by the field winding.
  • Therefore by the Faraday’s law of electromagnetic induction, EMF will be induced in the motor just like in the case of the DC generator.
  • So the question arises how the Back EMF comes from and why we are calling it as Back Emf?

Let’s get it easily.

As per Newton’s 3rd law which is applicable to electrical circuits as Lenz’s law says that every action opposes its cause. Similarly, in motors, the voltage applied is the cause for the motion and hence the applied voltage is opposed by the emf developed by the motor which is called as back emf.

  • In short back emf is generated by the generating action (moving conductor cutting the magnetic flux).
  • Therefore the back emf will have equation same as the emf equation of the generator.

EMF equation of motor

where Φ = Flux per pole
P = Number of pole
Z = Total number of conductors
A = Number of Parallel Path
N = Speed of a DC motor

  • The back emf is always less than the applied voltage “V” although the difference may be very small when the motor runs under its normal operating condition.


Why is DC motor self-regulating?

dc motor circuit diagram

Dc motor Circuit Diagram


  • In the given figure the DC motor is in series with the armature resistance Ra.
  • When the supply voltage is applied across the motor brushes the field magnets are excited and the electric current start to flow through the rotor armature, therefore, driving torque ” T ” is produced.
  • Due to this armature torque, the armature of DC motor rotates.
  • As the armature rotates the back Emf is generated in such a way that it tries to oppose the armature current which is produced by the supply voltage Va. 
  • So the voltage equation for the DC motor is
    Va = IaRa + Eb
  • Now from the back emf equation, we can say that the Eb is directly proportional to the speed of the DC motor so when the speed of DC motor reduces the back emf also reduces.
  • Since back Emf is smaller than the applied voltage, therefore, the difference between back emf and supply voltage increases i.e Va – Eb increases.
  • Now Ia = Va – Eb/ Rtherefore, armature current increases hence torque and speed increases.
  • From the above discussion, we can say that DC motor can maintain same speed with variable load.
  • Back emf act as governor i.e it makes motor self-regulating because it draws as much current as is just necessary.


Construction Of DC Generator

The principle of single loop DC generator has been described for the basic understanding of DC generator working. Now we will discuss the construction and working of actual generator.

The construction of DC generator and DC motor is same therefore instead of DC generator let say DC machine which is more appropriate and cover all aspect of DC motor and DC generator construction.

Direct current machine

construction of DC generator


A D.C machine consist of two main part:

  1. Stationary part
  2. Rotating part

Stationary part:

  • As the name signifies “stator” is the part which doesn’t move or remain fix in its position.
  • The stationary part of the DC machine consists of main poles.
  • The function of the main pole in DC machines is to provide a low reluctance path for the magnetic flux.
  • A low reluctance path provides the stronger magnetic field.
  • Other stationary parts of DC machine are frame/ yoke, field winding, pole shoe, etc.
  • So it is clear that the function of stationary part is to provide a magnetic flux.

Rotating part:

  • The rotating part of DC machine constituted of an armature core, armature windings, and a commutator.
  • It is also called as an armature of DC machine.
  • For generator mechacnical energy is converted into electrical energy.
  • For motor electrical energy is converted into mechanical energy.

Let’s discuss the various part of DC machine in detail.


  • The outer frame of a dc machine is called as the yoke.
  • The York serve two main purposes:
    It  provides mechanical strength to the whole assembly such as poles and acts as protecting cover for the whole machine.
    It carries the magnetic flux produced by the field winding.
  • The yoke can be constructed using cast iron or cast steel.
  • Cast iron is used for the smaller machine where cheapness is the main consideration.
  • Cast steel or rolled steel is used for large D.C machine.
  • The main disadvantage of using cast iron is that the cross-section of cast iron frame is twice that of the cast steel frame. Therefore DC machine becomes heavier in the case of cast iron.
  • The advantage of the Fabricated yoke is that the mechanical and magnetic properties of the are consistent.

Field coil

field coil of dc generator


  • A field coil is an electromagnet used to generate a magnetic field in an electromagnetic machine, such as motor and generator.
  • The field coil is also called as pole coil.
  • The field coil is consist of wounded copper wire or strip which carries the current to produce the desired flux.
  • Field Windings are wound all the poles with the particular current direction, and these windings are always connected in series.
  • This field winding produces alternate North-South pole. Since the Magnetic line of force flow from North pole to the South pole.

Pole Core

pole core & pole shoe


  • The Function of pole core is to carry the field winding.
  • The pole core serves two primary purposes:
  1. To support the field winding: Pole Core provides this area to wound the field winding
  2. To spread out the flux in air gap: Pole core direct the magnetic flux through the air gap, armature, and to the next pole.

So direction of flux is  YOKE ⇒ POLE ⇒ ARMATURE ⇒ NEXT POLE ⇒ YOKE

  • Pole core is either be made up of the solid piece of cast iron or cast steel or by the thin sheet of annealed steel lamination.
  • The thickness of lamination varies from 1 mm to 0.25 mm.

Pole shoe

  • Pole shoe is that part of the pole which will cover maximum armature conductors in its peripheral to cut the flux so that EMF induced will be more.
  • Pole shoe is laminated and is fastened to the pole face using counter sunk screws.

Armature core

  • The armature is the rotating part of DC machine.
  • Iron is used to build the armature core because of its excellent magnetic properties.
  • Eddy current in armature core is reduced by using lamination. So to reduce lamination core is made up of thin lamination.
  • Armature slots carry the armature conductor. Armature conductor or Armature winding is just like the field winding made up of copper.
  • Air duct in an armature core is provided for cooling purpose.
  • It should be noted that the EMF in the armature is alternating in nature in DC generator.

Armature winding

  •  An armature is that part of the DC machine where EMF is induced.
  • Armature coils are wound on the armature core and placed inside the armature slots.

Two methods can be used to wound armature coil:

  1. Lap winding
  2. Wave Winding
  • From this armature winding, we can take current or a voltage out of the DC machine.
  • Terminal box has two wire for input and two wires for outputs show in DC machine figure.
  • The output wire is coming from the armature conductor, therefore from this armature winding, we can take current or a voltage out of the DC machine.


  • The commutator is a mechanical rectifier, so the commutator collects induced EMF or current developed in the armature.
  • The commutator converts the alternating current generated in armature into the unidirectional current.
  • The commutator is made up of Copper segments, and each segment is insulated from each other by the thin layer of mica.
  • The number of commutator segments must be equal to the number of coils in the armature. That is, each commutator segment is in contact with two coil sides at all times.One coil side feeds the current to the segment, and the other side draws current from it.
  • Hence, the number of commutator segments is equal to twice the number of coil sides, which is equal to the number of coils.
  • Each commutator segment is connected to the armature conductor by using copper lugs.


  • We know that commutator is connected to the armature so as the armature is rotating (to cut the flux in order to induce the EMF ) obviously commutator will also rotate with the armature.
  • So in order to collect the current, we should have something which is stationarily and can be fit into the commutator, and that is brushes.
  • The number of brushes depends on how much current we need to tap from the commutator.
  • Brushes are made up material like carbon, copper, and graphite.
  • Copper brushes are used for the machine designed for large current at low voltage.
  • Graphite and Carbon Graphite are self-lubricated therefore widely used.

Armature shaft bearing

  • The ball bearing is used with the small machine.
  • For larger machine roller bearing are used with the driving end, and ball bearing is used with the non-driving (commutator) end.


Principle of DC Generator

Principle of DC Generator

A D.C generator is a machine which converts mechanical energy into electrical Energy. A D.C generator produces D.C power or current.  D.C generator is based on the principle of Faraday’s Law of Electromagnetic Induction.

Faraday’s Law of Electromagnetic Induction

  • Faraday’s law of electromagnetic induction describes the relationship between magnetic field and an electric circuit.
  • According to the Faraday’s law whenever a conductor is placed in the varying magnetic an E.M.F is induced in the conductor.
  • This induced E.M.F causes a current to flow in a closed path since in D.C generator is provided with a closed path.
  • The direction of induced E.M.F can be determined by Fleming’s right-hand rule.

From the above discussion, we can say that the essential condition of D.C generator are:-

  1. Magnetic field
  2. Conductors which can move as to cut the flux.

Before going further let us talk about some terms used in DC machine it will help you to understand the topic easily.

Difference between slip ring and split ring

Slip Ring

  • The slip ring is a continuous ring which is designed to make continuous contact between the fixed brush contacts.
  • The ring contacts on the shaft of a rotation object,  provide continuous power to items on the rotating shaft.
  • It is used in case of AC supply, where  AC is required.
  • With slip ring, the voltage in the external circuit varies like a sine wave and the current alternates the direction.

Split ring

  • Split ring is also called as commutator
  • A commutator has a ring with at least two breaks in it or can be divided into the number of segments.
  • The segments are insulated from each other by the thin sheet of Mica or any other insulating material.
  • An opposing pair of resulting contacts is wired to opposite poles of the motor.
  • A commutator not only allows current to flow but also allows for current reversal (synchronized with the rotation).
  • A split-ring commutator makes the current change direction every half-rotation.
  • The commutator is responsible for getting DC output although the internal voltage is AC.

So in short, slip rings are for continuous conduction and commutators are for synchronous reversals of the wiring.


When does Minimum and Maximum EMF is induced in the DC Generator?

This can be explained by Faraday’s Law of electromagnetic induction i.e  when a magnetic field is cut by moving conductor emf induced in it.

EMF is proportional to the rate of change of magnetic field lines that go through the coil.

Maximum induced EMF in an Alternator



Now from the above figure we can say that

  • When the plane of the coil is perpendicular it has lots of magnetic field lines passing through it.
  • The maximum EMF will be induced when the coil is perpendicular to the field.
  • When the plane of the coil parallel to the field and you have no field lines through the coil.
  • EMF is Zero when the coil is parallel to the field.

This can be explained by mathematical expression.

We know that

E = B x l x v
E =Blvsinθ

E is the generated voltage

B is the magnetic flux density
l is the length of a conductor in the magnetic field
v is the velocity of the conductor perpendicular to the magnetic lines.

The value of sinθ is maximum at 90 degrees i.e When the coil is perpendicular to the field axis.

The value of sinθ is minimum at 0 degrees i.e when the coil is parallel to the field axis.