The two circuits are said to
be coupled circuits if all or part of the electrical energy supplied to
one circuit is transferred to the other circuit without having any
electrical connection between them. Such coupled circuits are frequency
used in the circuit analysis. The common examples of the coupled
circuits are transformer, generator etc.
When the two circuits are placed very close to each other such that a
magnetic flux produced by one circuit links with both the circuits, then
the two circuits are said to be Magnetically Coupled Circuits. In this
article, we will consider magnetically coupled circuits consisting coils
or conductors. Such circuits are also called coupled inductors.
A wire of certain length, when twisted into coil becomes a basic
inductor. If a current is made to pass through an inductor, an
electromagnetic field is developed. A change in the magnitude of the
current, changes the electromagnetic field and hence induces a voltage
in coil according to Faraday’s law of electromagnetic induction.
When two or more coils are placed very close to each other, then the
current in one coil affects other coils by inducing voltage in them.
Such coils are said to be mutually coupled coils. Such induced voltages
in the coils are functions of the self inductances of the coils and
mutual inductance between them. Let us study the concepts of the self
inductance of a coil and mutual inductance.
Consider a coil having N turns carrying current i as shown in the Fig. 1.
|Fig. 1 Self Inductance|
flux ? is produced in the coil. The flux is measured in Wb (Weber). The
flux produced by the coil links with the coil itself. Thus the total
flux linkage of the coil will be (N? )Wb-turns. If the current flowing
through the coil changes, the flux produced in the coil also changes and
hence flux linkage also changes.
According to Faraday’s law, due to the rate of change of flux linkages,
there will be induced e.m.f. in the coil. This phenomenon is called
self induction. The e.m.f. or voltage induced in the coil due to the
change of its own flux linked with it, is called self induced e.m.f.
the self induced e.m.f. lasts till the current in coil is changing. The
direction of such induced e.m.f. is such that it opposes the cause
producing it i.e. change in current.
Thus when current increases, the self induced e.m.f. reduces the
current keeping it to its original value. And when current decreases,
the self induced e.m.f increases it to maintain current to its original
Key Point : Any change in current through the coil is opposed by the coil.
opposes any change in the current passing through it is called self
inductance of coil or only inductance of coil.
According to Faraday’s law of electromagnetic induction, self induced e.m.f. can be expressed as
V = -N (d?/dt) ……….(1)
The flux ? can be expressed as,
? = (Flux / Ampere) x Ampere = ?/i x i
Hence, Rate of change of flux = ?/i x Rate of change of current
d?/dt = ?/i . di/dt …………(2)
Putting value of d?/dt in equation (1), we can write,
V = -N(?/i . di/dt)
V = -(N?/i) di/dt ………(3)
The constant N?/i is called coefficient of self inductance and denoted by L.
L = N?/i ………(4)
Hence self induced e.m.f is given by
V= -L (di/dt) …………….(5)
Thus the magnitude of the induced e.m.f. is given by
V = L (di/dt) …………(6)
If the flux produced by one coil links with the other coil, placed
sufficiently close to the first coil, then due to the change in the flux
produced by first coil, there is induced e.m.f. in second coil. Such
induced e.m.f. in the second coil is called mutually induced e.m.f.
Due to the changing current in coil 1, the e.m.f. is induced in coil 2.
This phenomenon is called mutual induction and the induced e.m.f. is
called mutually induced e.m.f.
According to Faraday’s law, the magnitude of the induced e.m.f. is given by,
V2 = N2 (d?21/dt) ……………..(7)
Now, ?21 = ?21/i x i
If the permeability of the surroundings is assumed constant then ?21? i1 and hence ?21/i is constant.
... Rate of change of ?21 = ?21/i x Rate of change of i1
... d?21/dt = ?21/i1 . di1/dt ……………(8)
Putting value of d?21/dt in equation (7), the magnitude of the induced e.m.f. can be written by as,
V2 = N2 (?21/i1 . di1/dt)
... V2 = (N2 ?21/i1) di1/dt ……………(9)
Here the constant term (N2 ?21/i1)is defined as coefficient of mutual inductance and it is denoted by M.
... M = N2 ?21/ i1 ……………(10)
Then equation (9) can be written as,
... V2 = M di1/dt ………….(11)
The coefficient of mutual inductance is the property by which e.m.f.
gets induced in a coil because of change in current in other coil. It is
also called mutual inductance and it is measured in Henry (H).
Let us consider two coils having self inductance L1 and L2 placed close to each other as shown on the Fig.3.
This flux links with coil 2 as well as coil 1. So in each coil there
will be self induced e.m.f. as well as mutually induced e.m.f.
Let M be the mutual inductance between the two coils. Then the
magnitude of the self induced e.m.f. in coil 1 due to current i1 is L1 (di1/dt). The magnitude of the mutually induced e.m.f. in coil 1 due to current i2 in coil 2 is M di2/dt. Thus the magnitude of the total e.m.f. induced in coil 1 is given by,
V1 = L1 (di1/dt) + M(di2/dt) …………(12)
Similarly, in coil 2, there will be self induced e.m.f. due to current i2 and mutually induced e.m.f. due to current i1 in coil 1. The n the magnitude of the total e.m.f. induced in coil 2 is given by,
V2 = L2 (di2/dt) + M (di1/dt …………….(13)
Key Point : In a pair of linear
coupled circuits, a non-zero current in each coil produces mutually
induced voltage in other coil and self induced voltage in same coil. The
mutual inductance M is always positive but the mutually induced voltage
may be either positive or negative depending upon the reference current
directions and the physical construction of the coils.