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.
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