8.1. INDUCTION
When
D.C. is passed through an inductor (coil), a magnetic field is developed around
it account of magnetic effect of electric current. The field so produced has a
static nature. If A.C. is passed through the same inductor in place of D.C.,
the magnetic field too will have an alternating nature. And, an e.mf is
developed in a conductor placed in the alternating magnetic field. The
phenomenon is termed as induction.
Since,
the e.m.f. so produced has resulted in the static position of the conductor,
hence it is called statically induced e.m.f. The induction takes place in
accordance with the Faraday's law of electromagnetic induction (See sections
7.1 and 7.2 of Chapter 7),
According
to Lenz's law, the direction of induced e.m.f. at any instance is opposite to
the direction of applied e.m.f. Hence, the induced e.m.f. is also known as Back
e.m.f.
8.2. INDUCTANCE
(a)
Definition. The
property of a.c. circuits which opposes any change in the amount of current is
called inductance. Its symbol is L and its unit is henry (H).
(b) Henry. If a current,
changing at the rate of one ampere per second, induces an average e.m.f of one
volt in a conductor, the amount of inductance in the conductor will be one
henry.
8.3. SELF INDUCTANCE
The
property of one conductor or one circuit which opposes any, change in the
amount of current is called self inductance. Its symbol is L and its unit is
henry (H).
where, L = self inductance,
henrys
N = no. of turns
of the coil
= amount of flux passing through the
coil, Wb.
I =
current, amperes.
Since
the amount of induced e.m.f is directly proportional to the rate of change of
current
The
inductance of a solenoid (a coil whose length is about 10 times of its
diameter) can be calculated as follows:
where.
I. =
inductance of the coil, henrys
N = no. of
'turns
µ = permeability of the core
A = cross-sectional
area of coil, cm2
I = length of
the coil, cm.
The
inductance of a multilayered coil is calculated as follows:
where,
L = inductance of the coil, µH
N = no. of turns
a =
average radius of the coil, cm
b =
length of the coil, cm
c = total
thickness of the coil turns, cm.
Example 8.1.
Calculate the e.m.f. induced in a coil if a current changing at the rate of 50
mA per second is producing an inductance of 25 mH.
Example 8.2.
Calculate the inductance of a coil having 1200 turns and 4 cm, 2 cm, 4 cm size.
8.4. MUTUAL INDUCTANCE
The
properly of two circuits or windings (coils) which opposes any change in the
amount of current is called mutual inductance. Its symbol is M and its unit is
henry (H).
where,
M= mutual inductance, henrys
L1 =
self-inductance of firt coil, henrys
L2 =
self-inductance of second coil, henrys
8.5. COEFFICIENT OF COUPLING
If
two circuits or windings (coils) are arranged in such a manner that their
positions cause a transfer of electrical energy from one circuit to the other,
they are said to be coupled.
If
all the lines of force of the magnetic flux produced by the first coil pass
through the second coil, the coupling is said to be optimum, and its value is
referred as 100%. If only half or less no. of lines of force of the magnetic
flux produced by the first coil pass through the second coil, the coupling is
said to be loose and its value is referred as 50% or less as the case may be.
Hence,
the percentage of coupling is called coefficient of coupling.
where,
K = coefficient of
coupling
M = mutual
inductance, henrys
L1 = self-inductance of first coil,
henrys
L2
= self-inductance of second coil, henrys.
The
magnitude of K is always lesser than unity or 100%.
Example 8.3.
The inductances of two coupled coils are 10 mH and 20 mH respectively. If their
coefficient of coupling is 0.75 then calculate the mutual inductance of the two
coils.
8.6.
COUPLING IMPEDANCE
In
case of two coupled coils, the first coil induces an e.m.f. in the second coil
and similarly the second coil also induces an e.m.f. in the first coil. The
e.m.f. so induced in the first coil acts in such a direction so as to oppose
the applied e.m.f. In this way, the first coil has to bear an additional
impedance called coupling impedance.
The
effect of coupling impedance is the same as of connecting a resistor in series
with the first coil.
8.7. INDUCTIVE REACTANCE
The
opposition offered by an inductor to the flow of A. C. through it is called
inductive reactance. Its symbol is XL and its unit is ohm. The
amount of inductive reactance is directly proportional to the inductance and
the frequency.
where,
XL = inductive reactance, ohms
f =
frequency, Hz
L =
inductance, henrys.
Example 8.4.
Calculate the inductive reactance of a choke coil of 25 H working at 50 Hz
frequency. Also calculate circuit current at 200 volts.
Solution.
Given :
Inductance, L = 25 H
Frequency, f = 50 Hz
Voltage,
V = 200 volts
Hence, Inductive reactance, XL = 2
Ï€ . f. L.
= 2 x 3.14 x 50 x 25
= 3.14 x 25 x 100
= 7850 ohms
8.8. LAGGING OF CURRENT IN AN
INDUCTIVE CIRCUIT
As
the A.C. e.m.f. applied to a coil rises from zero to positive maximum, the
 |
| Fig. 8.1. Lagging of current in an inductive circuit |
back e.m.f. induced in the line wave representation coil also rises from zero to negative maximum. During this rise time the magnitudes of both the e.m.fs. remain almost equal, and therefore, the magnitude of current remains nearly zero. When both the e.m.fs. reach their peaks, they start to decrease. Meanwhile, the flow of current is started due to considerable difference between the applied and back e.m.fs. But, the current lags behind the voltage by 90°.
 |
| Fig 8.2. The angle of lag of current in an inductor |
The angle of lag of a pure inductive circuit is 90°. But, if the resistance of the circuit is also taken into consideration, the angle of lag is found to be less than 90°. The following formula is used for its calculation:
Example 8.5. If the inductive
reactance of a coil is 75 ohms and its resistance is 25 ohms then calculate the
angle of lag of current.
8.9. TIME CONSTANT
The
time taken by the current in reaching to 63.3% of its maximum value in a coil
is called its time constant. Its symbol is t and its unit is second.
where,
t = time
constant, seconds
L
= inductance, henrys
R = resistance, ohms.
Example
8.6. Calculate the time constant of a coil having an inductance of 400 mH and
resistance of 25 ohms.
Solution.
Given : Inductance, L = 400 mH
= 400 x 10-3 H
Resistance, R =
25 Ω 
8.10. INDUCTORS IN SERIES
Inductors
are connected in series for obtaining a high inductance value than that of a
single inductor. The magnitude of current remains the same throughout the
circuit and the voltage drop across inductors depends on the individual
inductance of inductors.
 |
| Fig. 8.3 Inductors in series |
In this way, the total inductance (LT) is equal to the sum of individual inductances (L1, L2, L3, ...) If there exists a coupling between the coils then —
where,
LT = total
inductance, henrys
L1, L2
= individual inductance, henrys
K = coefficient of coupling.
When
the two inductors are working in-phase then (+) sign is used and when they are
working out of phase then (–) sign is used.
Example 8.7.
Calculate the total inductance of two coils connected in series if their
inductances are 50 mH and 250 mH and the in phase coupling coefficient is 0.80.
8.11. INDUCTORS IN PARALLEL
Inductors
are connected in parallel for obtaining more output current. Inductors
connected in parallel get equal voltage and they draw currents as per their
individual inductances.
 |
| Fig 8.4 Inductors in parallel |
In
this way, the total inductance becomes lesser than that of the lowest
inductance value and the total circuit current is increased.
Example 8.8.
Two inductors of 4 and 8 henrys are connected in parallel. If the source
voltage is 200 volts, 50 Hz then calculate total inductance and total circuit
current.
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