Derive the expression of torque in closely excited system., Mechanical Engineering

Q. Derive the expression of torque developed in closely excited magnetic system. Clearly explain then assumption made.

 

Sol. Double - Excited System

 A doubly - excited magnetic system has two independent sources of excitations. Examples of such systems are separately excited dc machines synchronous machine, loudspeakers, tachometers etc.Let us consider that both the stator and rotor have silency. Assumptions are as for a singly - excited system.

 

       The flux linkage eq. for the two windings are

 

                                   Ψ1 = L1I1 + Mi2

 

                                   Ψ2 = L2I2 + Mi1

 

      The instantaneous voltage eq. for the two coils are

 

                                    V1 = R1i1 + d Ψ1/dt

 

                                    V2 = R2i2 + d Ψ2/dt

 

  Substituting the values Ψ1 and  Ψ2

 

                                    V1 = R1i1 + d/dt + (L1i1) + d/dt (Mi2)

 

                                     V2 = R2i2 + d/dt + (L2i2) + d/dt (Mi1)

 

Now the inductances are independent of currents and depend on the position of the root angle θm which is a function of time. Similarly, current are time dependent  and are not function of inductances. Therfore,

 

                                     V1 = R1i1 + L1di1/dt + i1dL1/dt + Mdi2/dt + i2dM/dt

 

                                     V2 = R2i2 + L2di12/dt + i2dL2/dt + Mdi1/dt + i1dM/dt

 

By multiplying we get,

 

                                    V1i1 = R1i1+ L1i1di1/dt + i12dL/dt + i1Mdi2/dt + i1i2dM/dt

 

                                    V2i2 = R2i2+ L2i2di2/dt + i22dL/dt + i2Mdi1/dt + i1i2dM/dt

 

Now we get,

 

                           (( v1i1 + v2i2 ) dt =(( R1i12 + R2i22 ) dt + (( L1i1di1 + L2i2di2 + i1Mdi2 + 2i1i2dM + i12dL1 + i22dL2 + i2Mdi1)

 

         Also,   [Useful electrical energy input] =  (( v1i1 + v2i2 ) dt - (( R1i12 + R2i22 ) dt

 

            [Energy to field storage in the electrical systems] + [Electrical to mechanical energy] = (( L1i1di1 + L2i2di2 + i1Mdi2 +2i1i2dM + i12dL1 + i22dL2 + i2Mdi1)

 

Stored energy in the Magnetic field

The instantaneous value of energy stored in the magnetic field depends on the inductance and current values at the instant considered. This energy may be found by considering the transductor to be stationary and the coils to be energized from zero current to the required instantaneous values of current. There is no mechanical output and Wem is zero. The inductance values are constant. Therefore terms dL1, dL2 and dM become zero

 

        (dWfe = oi1(L1i1di1 + oi2(L2i2di2 + oi2,i2 ( (i2Mdi1 + i1 Mdi2 )

 

         [Total Wfe] = 1/2L1i12 + 1/2L2i22 + Mi1i2

 

 Electromagnetic Torque

 

  If the transductor rotates, the rate of change of field energy with respect to time is given by differentiating.

 

             dWfe/dt = 1/2L1 d/dt i12 + 1/2i12 dL1/dt + 1/2L2 di2/dt2 + 1/2i22 dL2/dt + i2i2 dM/dt + i1M di2/dt + i2M di/dt

 

             dWfe/dt = L1i1 di1/dt  + 1/2i12 dL1/dt + L2i2 di2/dt + 1/2i22 dL2/dt + i2i2 dM/dt + i1M di2/dt + i2M di/dt

Integrated with respect to time

 

                       (dWfe = Wfe = ((L1i1di1 + 1/2i12dL1 + L2i2di2 + 1/2i22dL2) + i1i2dM + i1Mdi1

 

    This is general eq. for a moving transducer in which L1, L2 and M, i1 and i2 are all varying with position and time. On comparing we get,

 

           Wem = [Electrical to mechanical energy] =  ((1/2 i12dL1 + 1/2i22dL2 + i2i2dM)

 

               Differentiating with respect to θm

 

              dWem/d θm = ½ i12 dL1/d θm = ½ i22 dM/d θm

 

           as only L1, L2 and M are dependent on θm

 

       It includes the case of singly - excited system when one of the two current is equal to zero so that the expression for the torque becomes

 

                       Τe = i2/2 dl/d θm

 

         The first two terms of the torque are reluctance torques or saliency torques. The last term i1i2 dM/dθ is called the co - alignment torque, that is two superimposed fields, that try to align.

 

         For machines having uniform air gaps reluctance torque is not produced.

Posted Date: 7/23/2012 1:12:01 AM | Location : United States







Related Discussions:- Derive the expression of torque in closely excited system., Assignment Help, Ask Question on Derive the expression of torque in closely excited system., Get Answer, Expert's Help, Derive the expression of torque in closely excited system. Discussions

Write discussion on Derive the expression of torque in closely excited system.
Your posts are moderated
Related Questions
You are required, as part of a small team, to design a remote controlled buggy that can be made to move forwards, backwards and from side to side. The basic buggy and motors are pr

You have been selected to assist the structural engineer for a bridge replacement project.  Your task is to design the size of the footing for an interior bridge pier, based on the

Q. Design evacuation systems for plant? The plant will have designated Muster areas where personnel can go during an emergency and be accounted for and from which they can then

One end of a rod, 35 mm in diameter, is diped into a furnace with the other end projecting in the outer air. After the steady state is obtained, the temperature of the rod is evalu

State and explain parallelogram law of vectors

Define Cutting Parameters - Depth of Cut It is the thickness of metal removed by grinding wheel in one traverse stroke (lengthwise) t=depth of cut  =D 1 -D 2 / 2

Objectives After studying this unit, you should be able to explain various procedures in service and maintenance, and appreciate the importance of the inspections

A cantilevered beam is to be made of 7075-T6 aluminum with a uniaxial yield strength of 469 MPa. The beam is 1m long, and is loaded with the following limit loads: a bending mome

wire-frame entities-representing ellipse and parabolas

Magnitude and direction of force: Determine magnitude and direction of smallest force P which is required to start the wheel over the block. As shown in the figure. So