Open and closed position of the armature and flux linkage, Mechanical Engineering

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Q. For an electromagnetic system, show that the energy stored in a magnetic field is equal to the area enclosed b/w magnetization curves for open and closed position of the armature and flux linkage - current locus during the armature moment.

Sol. Consider a simple magnetic relay. Initially the armature is in the open position. When switch S is closed, current I is established in the N turn - coil. The flux set up depends upon mmf Ni and the reluctance of the magnetic path. The magnetic field thus produced, creates North and South poles and as a result of it., there is established a magnetic force tending to shortern the air - gap.

 

        If the armature is not allowed to move, the mechanical work done, dWmech. Is zero.

 

          Therefore,

 

               dWelec = 0 + dWfld

 

               (dWelec =   dWmech. + dWfld)

 

          This shows that when the movable part of any physical system is kept fixed, the entire electrical energy input is stored in the magnetic field.

 

                  dWfld = dWelec

 

and            dWfld = dWelec = i.dψ = F.dΦ

 

  If the initial flux is zero, then the magnetic field energy stored Wfld, in establishing a flux Φ1 or flux linkage ψ1, is given by

 

 

                  Wfld = ψ1 0(i.d ψ1 = Φ1 0(F.d Φ

 

I and F must be expressed in terms of ψ and Φ.

When the armature is held in open position then the most of the m.m.f. is consumed in the air - gap and it is likely that magnetic saturation may not occur.

 

             Wfld = Φ1 0( dWfld = Φ1 0(F.dQ = area OABO

 

            Wfld = ψ1 0(d. Wfld = ψ1 0(i.d ψ1 = area OABO

 

 

        Area OACO = (d. Wfld = F1 0( Φ.dF =   i1 0( ψ.di

 

         This area OACO is called the co - energy Wfld

 

W'fld = F1 0( Φ.dF =   i1 0( ψ.di

 

Above, ψ and Φ must be expressed in terms of F and I respectively. Co - energy has no physical significance, it is however useful in calculating the magnetic forces.

 

                        With no magnetic saturation,

 

                                   Area OABO = Area OACO

 

 Or                              Wfld =   W'fld

 

 And                     Wfld +  W'fld  = Area OCABO =  Φ1F1 = ψ1i1

 

In general, for magnetic circuit,

 

                                        Wfld =   W'fld  = 1/2 ψi = 1/2FΦ


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