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Lesson goals
In this lesson we shall show that (i) a time varying field will cause eddy currents to be
induced in the core causing power loss and (ii) hysteresis effect of the material also
causes additional power loss called hysteresis loss. The effect of both the losses will
make the core hotter. We must see that these two losses, (together called core loss) are
kept to a minimum in order to increase efficiency of the apparatus such as transformers &
rotating machines, where the core of the magnetic circuit is subjected to time varying
field. If we want to minimize something we must know the origin and factors on which
that something depends. In the following sections we first discuss eddy current
phenomenon and then the phenomenon of hysteresis.
Finally expressions for (i) inductance, (ii) stored energy density in a magnetic field and
(iii) force between parallel faces across the air gap of a magnetic circuit are derived.
Key Words: Hysteresis loss; hysteresis loop; eddy current loss; Faraday’s laws;
After going through this section students will be able to answer the following questions.
After going through this lesson, students are expected to have clear ideas of the
following:
1. Reasons for core losses.
2. That core loss is sum of hysteresis and eddy current losses.
3. Factors on which hysteresis loss depends.
4. Factors on which eddy current loss depends.
5. Effects of these losses on the performance of magnetic circuit.
6. How to reduce these losses?
7. Energy storing capability in a magnetic circuit.
8. Force acting between the parallel faces of iron separated by air gap.
9. Iron cored inductance and the factors on which its value depends.
22.2 Introduction
While discussing magnetic circuit in the previous lesson (no. 21) we assumed the exciting
current to be constant d.c. We also came to know how to calculate flux (φ) or flux density
(B) in the core for a constant exciting current. When the exciting current is a function of
time, it is expected that flux (φ) or flux density (B) will be functions of time too, since φ
produced depends on i. In addition if the current is also alternating in nature then both the magnitude of the flux and its direction will change in time. The magnetic material is now
therefore subjected to a time varying field instead of steady constant field with d.c
excitation. Let:
The exciting current i(t) = Imax sin ωt
Assuming linearity, flux density B(t) = μ0 μr H(t)
= μ0 μr
Ni
l
= sin max
0 r
N I ωt
μ μ l
∴ B(t) = BB
max sin ωt
22.2.1 Voltage induced in a stationary coil placed in a time varying field
If normal to the area of a coil, a time varying field φ(t) exists as in figure 22.1, then an
emf is induced in the coil. This emf will appear across the free ends 1 & 2 of the coil.
Whenever we talk about some voltage or emf, two things are important, namely the
magnitude of the voltage and its polarity. Faraday’s law tells us about the both.
Mathematically it is written as e(t) = -N d
dt
φ
Eddy current
Look at the Figure 22.3 where a rectangular core of magnetic material is shown along
with the exciting coil wrapped around it. Without any loss of generality, one may
consider this to be a part of a magnetic circuit. If the coil is excited from a sinusoidal
source, exciting current flowing will be sinusoidal too. Now put your attention to any of
the cross section of the core and imagine any arbitrary rectangular closed path abcd. An
emf will be induced in the path abcd following Faraday’s law. Here of course we don’t
require a switch S to close the path because the path is closed by itself by the conducting
magnetic material (say iron). Therefore a circulating current ieddy will result. The direction
of ieddy is shown at the instant when B(t) is increasing with time. It is important to note
here that to calculate induced voltage in the path, the value of flux to be taken is the flux
enclosed by the path i.e., φ max max =B × area of the loop abcd. The magnitude of the eddy
current will be limited by the path resistance, Rpath neglecting reactance effect. Eddy
current will therefore cause power loss in Rpath and heating of the core. To calculate the
total eddy current loss in the material we have to add all the power losses of different
eddy paths covering the whole cross section.
22.2.3 Use of thin plates or laminations for core
We must see that the power loss due to eddy current is minimized so that heating of the
core is reduced and efficiency of the machine or the apparatus is increased. It is obvious
if the cross sectional area of the eddy path is reduced then eddy voltage induced too will
be reduced (Eeddy ∞ area), hence eddy loss will be less. This can be achieved by using
several thin electrically insulated plates (called laminations) stacked together to form the
core instead a solid block of iron. The idea is depicted in the Figure 22.4 where the plates
have been shown for clarity, rather separated from each other. While assembling the core
the laminations are kept closely pact. Conclusion is that solid block of iron should not be used to construct the core when exciting current will be ac. However, if exciting current
is dc, the core need not be laminated.
Unidirectional time varying exciting current
Consider a magnetic circuit with constant (d.c) excitation current I0. Flux established will
have fixed value with a fixed direction. Suppose this final current I0 has been attained
from zero current slowly by energizing the coil from a potential divider arrangement as
depicted in Figure 22.7. Let us also assume that initially the core was not magnetized.
The exciting current therefore becomes a function of time till it reached the desired
current I and we stopped further increasing it. The flux too naturally will be function of
time and cause induced voltage e12 in the coil with a polarity to oppose the increase of
inflow of current as shown. The coil becomes a source of emf with terminal-1, +ve and
terminal-2, -ve. Recall that a source in which current enters through its +ve terminal
absorbs power or energy while it delivers power or energy when current comes out of the
+ve terminal. Therefore during the interval when i(t) is increasing the coil absorbs
energy. Is it possible to know how much energy does the coil absorb when current is
increased from 0 to I0? This is possible if we have the B-H curve of the material with us.