Mutually induced EMF and Its coefficient

1. Chandrashekhar S Patil
Sharad Institute of Technology
Polytechnic
Yadrav
Unit 1.5.1-Electric and Magnetic Circuits
(Mutually induced EMF and Its Coefficient)

2. Review of Previous Contents:-
Sr
No.
Topic
01. Induced EMF
02. Dynamically Induced EMF
03. Statically Induced EMF
04. Self Induced EMF
05. Magnitude and Coefficient of Self Induced EMF
2

3. Contents for current presentation
3
Sr No. Topic Slide Number
01. Factors affecting the inductance of a coil. 04.
02. Mutually Induced EMF. 05.
03. Magnitude of Mutually Induced EMF. 07.
04. Comparison between Self Induced EMF and
Dynamically Induced EMF.
09.
05. Coefficient of Mutual induction and Dimensions of
the coil
10.
06. Coefficient of coupling 11.

4. Factors affecting the Inductance of Coil
4
It is known that the inductance is proportional to the square of the number
of turns of the magnetizing coil and is inversely proportional to the
reluctance.
The reluctance in turn depends upon the dimensions of the coil (i.e. its
length and the cross sectional area) and the relative permeability of the
surrounding medium.
In this context following points are worth noting:
 Due to high relative permeability of iron an iron cored coil always has
more inductance than similar air cored coil(or coil with non magnetic
core).
 For air or other non magnetic materials relative permeability is always
constant (𝜇 𝑟=1)
Hence the inductance of a coil with the core of air or other non
magnetic material is
always constant and is independent of the value of current.
 In the case of an iron cored coil, the relative permeability of iron is not
constant but varies with the current due to saturation effect. As such
inductance of a coil also varies with the current.
 The inductance being dependent on number of turns the inductance of

5. Mutually Induced EMF
5
Consider two coils A and C placed side by side as shown
in the figure.
The coil A is connected in series with a switch(Sw), a
battery (B) and a variable resistor (R). A sensitive
galvanometer(G) is connected across the coil C.
Mutual Induction

6. 6
 On closing the switch let the current flowing in the coil A be I1 amperes.
Due to this current the coil A will have its own flux.
 Some of the flux produced by this coil will link with the coil C.If now the
current in the coil A is changed by varying R,It will change the flux linking
with the coil C.
 According to Faradays law this changing flux linking with the coil C will
induce an EMF in it. The production of the EMF will be indicated by the
deflection of the galvanometer.
 Thus any change of the current in the coil A will produce an emf in the
neighbouring coil C.This phenomenon is known as mutual induction.
 The EMF induced in the coil C in this manner is known as mutually
induced EMF or an EMF of mutual induction.
 The most important example of an appliance working on the principle of
mutual induction is transformer which is a static device used for changing
the magnitude of an alternating voltage
 Mutual inductance: The two magnetically coupled coils A and C in the
above mentioned example are said to have mutual inductance.
 Definition of Mutual Inductance: Defined as the property due to
which one coil (or circuit) with the change in its own current
produces an EMF in a nearby coil by induction.

7. Magnitude of Mutually Induced EMF
7
With reference to the previous figure:
N1=Number of turns of coil A.
N2=Number of turns of coil B.
I = Current flowing through coil A in amperes.
𝜑1=Flux produced by coil A due to current I1 in it.
𝜑2=Flux produced by coil B due to current I2 in it.
 The whole of the flux produced by coil A does not link coil C. Suppose a
fraction K1of this flux ie K1 𝜑1(where K1<1) is linked with coil C then 𝜑2=𝜑1
I1Webers.
 As already seen previously any change of the current in coil A changes
this flux and thereby produces an EMF in coil C by mutual induction. The
magnitude of this mutually induced EMF in coil C as given by equation
derived on Lenz law.
𝑒2= -N2
𝑑𝜑2
𝑑𝑡
-----------------------------------------
eqn 1.
Here the minus sign indicates that the induced EMF tends to circulate a
current in coil C in such a direction as to oppose the change of flux linking

8. 8
 Now , 𝜑2 =
𝜑2
I1
X I1---------------------------------eqn 2.
 But if the permeability of the surrounding medium is assumed to be
constant then 𝜑2 ∝ I1.
 But
𝜑2
I1
=Constant.
 Therefore Rate of change of 𝜑2=
𝜑2
I1
x Rate of change of I1.
 Therefore from equation 2.
𝑒2=-N2
𝜑2
I1
x Rate of change of I1
=-
N2K1 𝜑1
I1
x Rate of change of I1-----
eqn 3
The constant
N2K1 𝜑1
I1
in the above expression is called the coefficient of
mutual induction, coefficient of mutual inductance or simple mutual
inductance. It is the quantitative measure of mutual inductance and is
denoted by ‘M’.
Hence the coefficient of mutual inductance or coefficient of mutual
inductance (M)of two coils can be defined as the flux linkages (Weber-
turns) of one coil per ampere current in the other coil.
M=
N2K1 𝜑1
I1
henry.

9. 9

10. Coefficient of Mutual induction and Dimensions of the coil.
10
From Eqn 3. M=
N2K1 𝜑1
I1
But 𝝋 𝟏=
N 𝟏 𝑰 𝟏
𝒍/(µ𝒂)
=
N1I1
S
M=
N2K 𝟏
I 𝟏
X
N1I1
S
=
N1N2
S
henrys.
If all the flux produced by coil A Links with coil C.
K 𝟏=1.
M =
N1N2
S

11. Coefficient of coupling
11
The coefficient of coupling of the coils A and C can be defined as the ratio of
the actual mutual inductance present between the coils to its maximum
possible value

12. 12