Gandhinagar Institute Of Technology

Element Of Electrical Engineering (2110014)

Active Learning AssignmentTopic: Self Inductance, Mutual Inductance

AndCoefficient of Coupling

Guided By:

Branch: Mechanical Engineering Division: K

Prof. Rahish Silavat

Made By:

• Umang Shah (150120119178)• Saahil Kshatriya (150120119164)• Chintan (15012019167)

SELF INDUCTANCE

Self-InductanceWhen the switch is closed, the battery (source emf) starts pushing electrons around the circuit.

The current starts to rise, creating an increasing magnetic flux through the circuit.

This increasing flux creates an induced emf in the circuit.

The induced emf will create a flux to oppose the increasing flux.

The direction of this induced emf will be opposite the source emf.

This results in a gradual increase in the current rather than an instantaneous one.

The induced emf is called the self-induced emf or back emf.

•When an electric current is passed through an insulated conducting coil, it gives rise to a magnetic field in the coil so that the coil itself behaves like a magnet.

•The magnetic flux produced by the current in the coil is linked with the coil itself.

DEFINATION

As the strength of the current in thecoil is changed, the flux linked with the coilalso changes. Under such circumstances anemf is induced in the coil too. Such emf iscalled a self-induced emf and this phenomenon is known as self-induction.

Conducting coil

Battery Key/Switch Rheostat

Wires

Direction of the Current

Current flows In anti-clock Wise direction

Current flows In clock Wise direction

If the number of turn in a coil is N and the flux linked with each turn is φ, then the total flux linked through the coil = Nφ. In this case, the total flux linked with the coil (which is called flux linkage) is directly proportional to the current I flowing through the coil.

N = LI

where the constant of proportionality L is called the self-inductance of a coil.

N = LI, L= N/I

The self inductance L is a measure of the flux linked with coil per unit current.

•The self-inductance L of a coil depends upon –

(1)The size and shape of the coil.

(2) The number of turns N.

(3) The magnetic property of the medium within the coil in which the flux exists.

NOTE:Self-inductance L does not depend on current I.

Diffrentiating equation N = LI with respect to time t,

N d/dt = L dI/dt

In the case of self-induction, Faraday’s law and Lenz’s law holds good. Hence self-induced emf in the coil is,

e = -N d/dt

Self-induced emf is also called “back emf”.

Form equation e = -L dI/dt

Self inductance L = -e/(dI/dt)

“The self-induced emf produced per unit rate of change of current in the circuits called self-inductance of the circuit.”

Unit of L =unit of emf(v)/Unit of rate of change of current (A/s)=Vs/A or Henry(H)

Mutual Induction

Mutual induction is the phenomenon of production of induced emf in one coil due to a change of current in the neighboring coil.

Coefficient Of mutual inductionAt any instant ,Magnetic Flux linked with the secondary coil current in the primary coil.∝

i.e. ᵩ I∝

ᵩ =MI

The proportionality constant M is called th mutual inductance or coeffecient of mutual induction of the two coils.Any change in the current I sets up and induced emf in the secondary coil which is given by

E= = -M

Consider two long co-axial solenoids S1 and S2, with S2 wound over S1 .

Let l= length of each solenoid r1, r2=radii of the two solenoids = area of cross section of inner solenoid S1 N1, N2 = number of turns in the two solenoidsFirst we pass a time varying current I2 through S2 . The magnet field setup inside S2 due to I2

B2 = µ0 n2 I2

where n2 = N2 /l =number of turns per unit length of S2.

Total magnetic flux linked with the inner solenoid S1 isΦ1 = B2 A N1 = µ0 n2 I2 A N1 Mutual inductance of coil 1 w.r.t coil 2 is M12 = Φ1 / I2 = µ0 n2 A N1 = µ0 N2 A N1 / l

We now consider the flux linked with the outer solenoid S2 due to the current I1 in the inner solenoid S1 . The field B1 due to I1 is constant inside S1 but zero in the annular region between the two solenoids. hence.B1 = µ0 n1 I1 Where n1 = N1 /l = the number of turns per unit length of S1 The total flux linked with the outer solenoid S2 isΦ2 = B1 A N2 = µ0 n1 I1 A N2 = µ0 N2 A N1 I1 / l Mutual inductance of coil 2 w.r.t coil 1 isM21 = Φ2 / I1 = µ0 N2 A N1 / lClearly, M12 = M21 = M

M= µ0 N2 A N1 / l

Thus mutual inductance of two coils is the property of their combination. It does not matter which one of them functions as primary or secondary coil.

Factors on which mutual inductance depends

1. Number of Turns : larger the number of turns , larger the mutual inductance.

2. Larger the common cross-sectional area of two solenoids , larger will be their mutual inductance.

3. Larger the distance b/w two solenoids , smaller will be the magnetic flux linked with the secondary coil due to current in the primary coil. Hence smaller will be the value of M.

Coefficient Of

Coupling

Consider two coils having self inductance L1 and L2 placed very close to each other. Let the number of turns of the two coils be N1 and N2 respectively. Let coil 1 carries current i1 and coil 2 carries current i2.

Due to current i1, the flux produced is Φ1 which links with both the coils. Then from the previous knowledge mutual inductance between two coils can be written as M = N1 Φ21/i1 ...............(14) where Φ21 is the part of the flux Φ1 linking with coil 2. Hence we can write, Φ21 = k1 Φ1.

... M = N1 ( k1 Φ1)/i1 .................(15)

Similarly due to current i2, the flux produced is Φ2 which links with both the coils. Then the mutual inductance between two coils can be written as M=N2 Φ21/i2 .........(1) where Φ21 is the part of the flux Φ2 linking with coil 1. Hence we can write Φ21 = k2 Φ2.... M=N2 (k2 Φ2)/i2 ..................(2)

Multiplying equations (15) and (17),

But             N1Φ1/i1 = Self induced of coil 1 = L1

                   N2Φ2/i2 = Self induced of coil 2 = L2

...                M2 = k1k2L1L2

...                 M = √(k1k2) √(L1L2)Let                k = √(k1k2) 

...                   M = k √(L1L2)                            ............(18)      where k is called coefficient of coupling....                    k = M/(√(L1L2))                                .........(19)

THANK YOU

The coefficient of coupling gives idea about the magnetic coupling between the two coils. So when the entire flux in one coil links with the other, the coupling coefficient is maximum. The maximum value of k is unity. Thus when k = 1, the coupled coils are called tightly or perfectly coupled coils. Also the mutual inductance between the two coils is maximum with k =1. The maximum value of the mutual inductance is given by M = √(L1L2)