ELECTRICAL MACHINES Electrical Machines

CHAPTER 1 ELECTROMECHANICAL ENERGY CONVERSION Electrical Machines

Introduction Electro mechanical device converts energy between electrical and mechanical forms. Electromechanical energy conversion theory helps in representing electromagnetic force or torque in terms of such parameters like electric current or displacement. Electrical Machines

Expression for Induced E.M.F. The magnitude of e.m.f. generated will depend on the number of turns in the coil, strength of the magnetic field and the speed of rotation of the coil or the magnetic field. Electrical Machines

Expression for Induced E.M.F. Let us consider a multi turn coil of length ‘l’ meters, width ‘b’ meters having ‘T’ turns is rotated at an angular speed inside a magnetic field. Let, ‘n’ is the speed of the coil in r.p.s. and B be the flux density in the air gap. Electrical Machines

Expression for Induced E.M.F. Then the e.m.f. induced in a multi turn coil will be, The direction of induced e.m.f. is obtained by “Flemings left hand rule”. Electrical Machines

Force and Torque in a Magnetic Field The force acting on a moving particle having a charge q, when placed in a magnetic field is given by the Lorentz`s force law. Where, F is the force in Newton, q is the charge in coulomb, B is the flux density in Tesla, Electrical Machines

Force and Torque in a Magnetic Field E is the volts per meter and v is the velocity of the particle in meters per sec. When large numbers of charged particle are in motion then the equation becomes Electrical Machines

Energy Balance Concept The law of principle of conservation of energy states that, energy cannot be created or destroyed but it possible to transform energy from one form to the some other form. Electrical Machines

Energy Balance Concept Thus the energy balance equation For motoring action: Energy input = Mechanical output + Energy loss + Total energy stored. For generating action: Mechanical input = Electrical output + Energy loss + Total energy stored. Electrical Machines

Laws of electromagnetism There are four laws of electromagnetism Faraday`s Law. Lenz`s Law The Biot-Savart law Ampere's law Electrical Machines

Faraday`s Laws of Electromagnetic Induction Faraday`s First Law: When the magnetic flux linking with the coil changes, an e.m.f. is induced in the coil. Faraday`s Second Law: The magnitude of induced e.m.f. is the rate of change of flux linkage in it. Electrical Machines

Lenz’s Law It states that the direction of induced e.m.f. in the conductor will always oppose the cause of its production. The negative sign in the above equation signifies this effect. Electrical Machines

The Biot-Savart Law Any current element projects into space a magnetic field, the magnetic flux density of which dB, is directly proportional to the length of the element dl, the current I, the sine of the angle and θ between direction of the current and the vector joining a given point of the field and the current element and is inversely proportional to the square of the distance of the given point from the current element, r. Electrical Machines

Ampere’s Law Ampere's law states that for any closed loop, the sum of the length elements times the magnetic field in the direction of the length element is equal to the permeability times the electric current enclosed in the loop. Electrical Machines

Induced E.M.F. Induced emf Dynamically induced e.m.f. Statically induced e.m.f Mutually induced e.m.f. Self induced e.m.f. Electrical Machines

Self Induced e.m.f. Self inductance is the phenomenon in which if the current flowing through the inductive coil changes, then an induced e.m.f. will be produced in the coil. Electrical Machines

Mutually induced e.m.f. It is the phenomenon in which a change of current in one coil causes an induced e.m.f. in another coil placed near to the first coil. Electrical Machines

Energy Stored in a Magnetic Field Consider a coil with N turns wound on a magnetic material. This magnetic material will act as a core. When the coil is supplied with a voltage of volts, current i will flow through it. Let R be the resistance of the coil. Then from power equation will be Electrical Machines

Energy Stored in a Magnetic Field Electrical Machines

Singly Excited Linear Actuator Electrical Machines

Singly Excited Linear Actuator Let us consider a linear actuator as shown on Slide 21. A coil having resistance R is wound on a magnetic core. When this coil is excited with a voltage current i will flow through the coil. Let us assume that after some time the movable plunger is moved by a distance under the action of force F. Then, the mechanical work done will be, dWm =Fdx Electrical Machines

Singly Excited Linear Actuator The electrical energy that has been transferred into the magnetic field and also converted into mechanical work will be, Electrical Machines

Singly Excited Linear Actuator Electrical Machines

Singly Excited Linear Actuator The force acting on the plunger will be, Where, L is the self inductance of the coil. Electrical Machines

Singly Excited Linear Actuator The area OABO is called the co-energy. Co-energy has no physical significance but it is important in obtaining the magnetic force. Electrical Machines

Single Excited Rotating System Electrical Machines

Single Excited Rotating System A coil having resistance R1 is wound on a magnetic core. When this coil is excited with a voltage v1 current i1 flows through the coil. Let us assume that all the flux produced by the coil will pass through the core, that is, there is no leakage flux. Electrical Machines

Single Excited Rotating System The movable part will experience torque and will try to position itself in such a position, where minimum reluctance is given to the magnetic flux. Electrical Machines

Single Excited Rotating System Electrical Machines

Single Excited Rotating System Electrical Machines

Single Excited Rotating System Area OCDO and OBAO is the stored energy when the movable rotor is in position-1 and position-2 respectively. Let us assume that the rotor is initially positioned at position-1 (operating point C in the Fig.1.6). Electrical Machines

Single Excited Rotating System At this position let the current through the circuit be i1. Let us assume that the rotor is moving very slowly towards position-2. Since the movement of the rotor is very slow, the current through the circuit will remain constant (i1). When the rotor will reach position-2, the flux linkage will change. Electrical Machines

Single Excited Rotating System Electrical Machines

Double Excited System Electrical Machines

Double Excited System The stator is fed from a voltage source of v1 because of which current i1 is flowing through it. Let R1 be the resistance of the stator winding. Similarly, the rotor is fed from a voltage source of v2 because of which current i2 is flowing through it. Electrical Machines

Double Excited System Let R2 be the resistance of the rotor winding. Let the self inductance of the stator and the rotor be L1 and L2 respectively and M be the mutual inductance between them. The flux linkage equations for the two windings are Electrical Machines

Double Excited System Electrical Machines

Double Excited System Solving we get, Electrical Machines

Dynamic Equation of Electromechanical Systems Let us consider a simple transducer system Electrical Machines

Dynamic Equation of Electromechanical Systems This electromechanical system can be divided into three parts. Electrical Machines

Dynamic Equation of Electromechanical Systems From the circuit we get. Electrical Machines

The Torque Equation of a Machine with Cylindrical Air Gap Electrical Machines

The Torque Equation of a Machine with Cylindrical Air Gap This system can be considered of having two windings, the stator winding and the rotor winding. The stator is generally the stationary part of the system and the rotor is the rotating part of the system. In A.C. machines both the stator and the rotor windings carry currents and thus produces their own magnetic fields. Electrical Machines

The Torque Equation of a Machine with Cylindrical Air Gap Torque is produced due to the tendency of these two magnetic fields to line up in the same direction, which in turn makes the rotor to rotate. Voltage is produced due to the relative motion between the two windings. Electrical Machines

The Torque Equation of a Machine with Cylindrical Air Gap Torque equation is The generalized torque equation for machines having P poles will be, Electrical Machines