## Introduction to Coupled Circuits

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.

__1.1 Self 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.

**Key Point**:

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

value.

**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)

**Key point**: The coefficient of self inductance is defined as the flux linkages per ampere current in it. Its unit is Henry (H).

__1.2 Mutual Inductance__

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.

_{1 }turns, while coil 2 has N

_{2 }turns. The current flowing through coil 1 is i

_{1}. Due to this current, the flux produced in coil 1 is ?

_{1}. The part of this flux links with coil 2. This flux is called mutual flux.

_{21 }a sit is a part of flux ?

_{1}linking with coil 2. When current through coil 1 changes, the flux produced in coil 1 i.e. ?

_{1 }changes. Thus flux associated with coil 2 i.e. ?

_{21}changes. So according to the Faraday’s law, there will be induced e.m.f. in coil 2.

**Key Point**:

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,

V_{2 } = N_{2 }(d?_{21}/dt) ……………..(7)

Now, ?_{21 } = ?_{21}/i x i

If the permeability of the surroundings is assumed constant then ?_{21}? _{ }i_{1 }and hence ?_{21}/i is constant.

**. ^{.}. **Rate of change of ?

_{21}= ?

_{21}/i x Rate of change of i

_{1 }

**.**d?

^{.}._{21}/dt = ?

_{21}/i

_{1 }

**.**di

_{1}/dt ……………(8)

Putting value of d?

_{21}/dt in equation (7), the magnitude of the induced e.m.f. can be written by as,

V

_{2}= N

_{2 }(?

_{21}/i

_{1 }

**.**di

_{1}/dt)

**.**V

^{.}._{2}= (N

_{2 }?

_{21}/i

_{1}) di

_{1}/dt ……………(9)

Here the constant term (N

_{2 }?

_{21}/i

_{1})is defined as coefficient of mutual inductance and it is denoted by M.

**.**M = N

^{.}._{2 }?

_{21}/ i

_{1}……………(10)

Then equation (9) can be written as,

**.**V

^{.}._{2}= M di

_{1}/dt ………….(11)

**Key Point**:

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 L_{1} and L_{2} placed close to each other as shown on the Fig.3.

Fig. 3 |

_{1}; the flux produced is ?

_{1}. This flux links with coil 1 and coil 2 . Similarly due to current i

_{2}, the flux produced is ?

_{2}.

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 i_{1} is L_{1} (di_{1}/dt). The magnitude of the mutually induced e.m.f. in coil 1 due to current i_{2} in coil 2 is M di_{2}/dt. Thus the magnitude of the total e.m.f. induced in coil 1 is given by,

V_{1} = L_{1} (di_{1}/dt) + M(di_{2}/dt) …………(12)

Similarly, in coil 2, there will be self induced e.m.f. due to current i_{2 }and mutually induced e.m.f. due to current i_{1 }in coil 1. The n the magnitude of the total e.m.f. induced in coil 2 is given by,

V_{2} = L_{2} (di_{2}/dt) + M (di_{1}/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.

## Leave a comment