26. Explain the basic purpose of a tertiary winding. To what additional use can it be put?
4.2 ENERGY IN MAGNETIC SYSTEM
4.2 ENERGY IN MAGNETIC SYSTEM
Energy can be stored or retrieved from a magnetic system by means of an exciting coil connected to an electric source. Consider, for example the magnetic system of an attracted armature relay of Fig. 4.1. The resistance of the coil is shown by a series lumping outside the coil which then is regarded as an ideal loss-less
coil. The coil current causes magnetic flux to be established in the magnetic circuit. It is assumed that all the flux f is confined* to the iron core and therefore links all the N turns creating the coil flux linkages of
l = Nf (4.1) The flux linkage causes a reaction emf of
e = d dt
l (4.2)
to appear at the coil terminals with polarity (as per Lenz’s law) shown in the Fig. 4.1. The associated circuit equation is
v = iR + e
= iR + d
dt
l (4.3)
The electric energy input into the ideal coil due to the flow of current i in time dt is
dWe = ei dt (4.4)
Assuming for the time being that the armature is held fixed at position x, all the input energy is stored in the magnetic field. Thus
dWe = ei dt = dWf (4.5)
where dWf is the change in field energy in time dt. When the expression for e in Eq. (4.2) is substituted in Eq. (4.5), we have
dWe = idl = F df = dWf (4.6)
where F = Ni, the magnetomotive force (mmf ).
The relationship i-l or F-l is a functional one corresponding to the magnetic circuit which in general is nonlinear (and is also history-dependent, i.e. it exhibits hysteresis). The energy absorbed by the field for finite change in flux linkages for flux is obtained from Eq. (4.6) as
DWf =
As the flux in the magnetic circuit undergoes a cycle f1Æf2Æf1, an irrecoverable loss in energy takes place due to hysteresis and eddy-currents in the iron, assuming here that these losses are separated out and are supplied directly by the electric source. This assumption renders the ideal coil and the magnetic circuit as a conservative system with energy interchange between themselves so that the net energy is conserved.
The energy absorbed by the magnetic system to establish flux f (or flux linkages l) from initial zero flux is Wf =
0
Ú
l i(l) dl = 0Ú
f F(f) df (4.8)This then is the energy of the magnetic field with given mechanical configuration when its state corresponds to flux f (or flux linkages l).
* The leakage flux (which is of course small) does not take part in the energy conversion process. It can be ac-counted for by placing an imaginary coil in series with the ideal coil which produces exactly the flux linkages corresponding to the leakage flux. As in the case of transformers, the inductance of such a coil is referred to as the leakage inductance. Here the leakage inductance is assumed to be negligible.
Core Fluxf x
Fig. 4.1 Attracted armature relay
The i-l relationship is indeed the magnetization curve which varies with the configuration variable x (Fig. 4.1: the air-gap between the armature and core varies with position x of the armature. The total reluctance of the magnetic path decreases as x increases). The i-l relationship for various values of x is indicated in Fig. 4.2. It immediately follows that this relationship can be expressed as
i = i(l, x) if l is the independent variable or as
l = l(i, x) if i is the independent variable.
Therefore, the field energy (Eq. (4.8)) is in general a function of two variables,
i.e. Wf = Wf (l, x) (4.9a)
or Wf = Wf (i, x) (4.9b)
According to Eqs (4.9a) and (4.9b) field energy is determined by the instantaneous values of the system states ((l, x) or (i, x) and is independent of the path followed by these states to reach the present values. This means that the field energy at any instant is history independent.
A change in l with fixed x causes electric-magnetic energy interchange governed by the circuit Eq. (4.3) and the energy Eq. (4.6). Similarly, if x is allowed to change with fixed l, energy will interchange between the magnetic circuit and the mechanical system. The general case of such energy interchanges (electric-magnetic-mechanical) is the subject matter of Sec. 4.3.
As per Eq. (4.8) the field energy is the area between the l-axis and i-l. curve as shown in Fig. 4.3. A new term, co-energy is now defined as
W¢f (i, x) = il – Wf (l, x) (4.10)
wherein by expressing l as l(i, x), the independent variables of W ¢f become i and x. The coenergy on Fig. 4.3 is shown to be the complementary area of the i-l curve. It easily follows from Fig. 4.3 that
W¢f = il di
Ú
0 (4.11)Wf= field energy
W'f= coenergy i- curve for fixedl x l-axis
l
i-axis
0 i
Fig. 4.3 Field energy and coenergy l
0 i
x1 x2 x3
x1>x2>x3
Fig. 4.2 i -l relationship with variable x
Linear Case
Electromechanical energy conversion devices are built with air-gaps in the magnetic circuit which serve to separate the stationary and moving members. As a result the i-l relationship of the magnetic circuit is almost linear; also the losses of magnetic origin are separately accounted for by semi-empirical methods. With the linearity assumption the analysis is greatly simplified. Losses and certain nonlinear effects may then be incorporated at a later-stage.
Assuming linearity, it follows from Eq. (4.8) or Fig. 4.3 that Wf = 1
2il = 1
2Ff = 1
2Rf2 (4.12)
where, as it is known, R = F/f = reluctance of the magnetic circuit. Since the coil inductance L = l/i
the field energy can be expressed as
Wf = 1 2 l2
L (4.13)
In the linear case the inductance L is independent of i but is a function of configuration x. Thus the field energy is a special function of two independent variables l and x, i.e.
Wf (l, x) = 1 2
l2
L x( ) (4.14)
The field energy is distributed throughout the space occupied by the field. Assuming no losses and constant permeability, the energy density* of the field is
wf =
where H = magnetic field intensity (AT/m) B = magnetic flux density (T)
The energy density expression of Eq. (4.15) is important from the point of view of design wherein the capability of the material is to be fully utilized in arriving at the gross dimensions of the device.
For the linear case it easily follows from Eq. (4.11) that coenergy is numerically equal to energy, i.e.
W¢f = Wf = 1 2li = 1
2Ff = 1
2PF2 (4.16)
where P = f/F = permeance of the magnetic circuit.
Also in terms of the coil inductance W¢f =
The expression for coenergy density is w¢f =
0
Ú
H B dH (4.18a)which for the linear case becomes w¢f = 1
2mH2 =1 2
B2
m (4. 18b)