Test: Mutual Inductance - Electrical Engineering (EE) MCQ

Concept:

The mutual inductance is given by:

where, M = Mutual inductance

L = Self-inductance

k = Coefficient of coupling

The value of k lies between 0 and 1.

The maximum value of mutual inductance is possible for k = 1.

Calculation:

Given, L₁ = 10 mH

L₂ = 40 mH

M = 20 mH

Concept:

The equivalent inductance of series aiding connection is

L_eq = L₁ + L₂ + 2M

The equivalent inductance of series opposing connection is

L_eq = L₁ + L₂ - 2M

Calculation:

Given:

L₁ = 20 mH

L₂ = 40 mH

L_eq = 50 mH

From the given data we can conclude that series opposing connection will be considered as the mutual index cannot be negative.

L_eq = L₁ + L₂ - 2M

50 = 40 + 20 - 2M

M = 5 mH

Concept:

Coefficient of Coupling (k):

The coefficient of coupling (k) between two coils is defined as the fraction of magnetic flux produced by the current in one coil that links the other.

Two coils have self-inductance L1 and L2, then mutual inductance M between them then Coefficient of Coupling (k) is given by 

N1 and N2 is the number of turns in coil 1 and coil 2 respectively

A is the cross-section area

l is the length

Calculation:

Given,

L1 = 75 mH

L2 = 105 mH

k = 0.45 

From the above concept,

M = 39.93 mH

 Whenever more than one inductors come in closer, there may be mutual induction between them.

If more than one inductor is connected in series and the flux of one inductor links the other, we have to consider mutual inductance during equivalent inductance calculations.

If a circuit contains two inductors of self-inductance L₁​ and L₂​ in series as shown in the figure. If M is the mutual inductance, then find the effective inductance of the circuit shown.

L_eq = L₁ + L₂ + 2M

Mutual Inductance:

• The mutual inductance of two coils is defined as the emf induced due to the magnetic field in one coil opposing the change of current and voltage in another coil. 
• Thus mutual inductance depends on the value of current in the circuit.
• That means the two coils are magnetically linked together due to the change in magnetic flux.
• The magnetic field or flux of one coil links with another coil. This is denoted by M.

• The above figure shows that mutual inductance is a geometric quantity.
• The current flowing in one coil induces the voltage in another coil due to the change in magnetic flux.
• The amount of magnetic flux linked with the two coils is directly proportional to the mutual inductance and current change.

Mutual inductance is given by 

Where μ₀ = Permeability of free space = 4π × 10^-7 H/m

μ_r = Relative permeability of the core

N₁ = Number of turns of coil 1

N₂ = Number of turns of coil 2

A = Cross-sectional area in m²

• The mutual inductance of two coils can be increased by placing them on a soft iron core or by increasing the number of turns of the two coils.
• Unity coupling exists between the two coils when they are tightly wound on a soft iron core. The leakage of flux would be small.
• If the distance between the two coils is short, then the magnetic flux produced in the first coil interacts with all the turns of the second coil, which results in large EMF and mutual inductance.

• If the two coils are farther and apart from each other at different angles, then the induced magnetic flux in the first coil generates weak or small EMF in the second coil.
• Hence the mutual inductance will also be small.

• Thus the value of this mainly depends on the positioning and spacing of two coils on a soft iron core.
• Consider the figure, which shows that the two coils are tightly wound one on the top of the soft iron core.

• The change of current in the first coil produces a magnetic field and passes the magnetic lines through the second coil, which is used to calculate mutual inductance.
• A transformer works on the principles of "electromagnetic induction" as a mutual induction i.e. transformers have a large mutual inductance.
• Mutual inductance is the main operating principle of generators, motors, and transformers.
• Any electrical device having components that tend to interact with another magnetic field also follows the same principle.
• The interaction is usually brought about by a mutual induction where the current flowing in one coil generates a voltage in a secondary coil.

Tesla Coil

• A Tesla coil is an electrical resonant transformer circuit designed by inventor Nikola Tesla in 1891.
• It is used to produce high-voltage, low-current, high-frequency alternating-current electricity.
• The Tesla coil works on a principle to achieve a condition called resonance. Here, the primary coil emits huge amounts of current into the secondary coil to drive the secondary circuit with maximum energy.
• The fine-tuned circuit helps to shoot the current from the primary to the secondary circuit at a tuned resonant frequency.
• It takes the output from a 120vAC to several kilovolt transformers and driver circuits and steps it up to an extremely high voltage.
• Voltages can get to be well above 1,000,000 volts and are discharged in the form of electrical arcs.

Additional Information 

• Mutual inductance occurs when one magnetic field interacts with another.
• The mutual inductance concept governs the operation of transformers, motors, generators, and other electrical equipment.
• Mutual induction occurs when a current flows through one coil or winding and induces a voltage in another coil.

Concept: 

The equivalent inductance of two inductors in a series connection can be calculated as

For mutual aiding (when polarity dots are present at the same ends)

For mutual opposing (when polarity dots are present at the opposite ends)

Calculation:

For series aiding, the equivalent inductance is 1.4 mH, i.e.

For series opposing, the equivalent inductance is 0.6 mH, i.e.

Subtracting the two equations, we get:

1.4 mH - 0.6 mH = 4 M

4 M = 0.8 mH

M = 0.2 mH = 0.2 × 10^-3 Henry

Concept:

Self-inductance is the property of the current-carrying coil that resists or opposes the change of current flowing through it. This occurs mainly due to the self-induced emf produced in the coil itself. In simple terms, we can also say that self-inductance is a phenomenon where there is the induction of a voltage in a current-carrying wire.

Self-inductance, usually just called inductance, L is the ratio between the induced voltage and the rate of change of the current

V₁(t) = L (di₁/dt)

Mutual Inductance between the two coils is defined as the property of the coil due to which it opposes the change of current in the other coil, or you can say in the neighboring coil. When the current in the neighboring coil changes, the flux sets up in the coil and, because of this, changing flux emf is induced in the coil called Mutually Induced emf and, the phenomenon is known as Mutual Inductance.

V₂(t) = M (di₁/dt)

Coefficient of Coupling:

The amount of coupling between the inductively coupled coils is expressed in terms of the coefficient of coupling, which is defined as

where M = mutual inductance between the coils

L₁ = self-inductance of the first coil, and

L₂ = self-inductance of the second coil

• The coefficient of coupling is always less than unity and has a maximum value of 1 (or 100%).
• This case, for which K = 1, is called perfect coupling when the entire flux of one coil links the other.
•  The greater the coefficient of coupling between the two coils, the greater the mutual inductance between them, and vice-versa.

Calculations:

Given 

L₁ = 0.5 H

L₂ = 4 H

K = 0.8

I = 20 sin 314t

The induced emf in coil 2 due to current in coil 1 is given by

V₂ = M (di₁ / dt)

M = 1.13 H

V₂ = 1.13 d/dt (20 sin 314t)

V₂ = 1.13 ×  20 × 314 × cos 314t

The maximum value of V₂ = 1.13 × 20 × 314 = 7.1 kV

Concept:

Coefficient of Coupling (k):
The coefficient of coupling (k) between two coils is defined as the fraction of magnetic flux produced by the current in one coil that links the other.

Two coils have self-inductance L₁ and L₂, then mutual inductance M between them then Coefficient of Coupling (k) is given by 

N₁ and N₂ is the number of turns in coil 1 and coil 2 respectively
A is the cross-section area
l is the length

Calculation:

Given,

L₁ = 4mH

L₂ = 9mH

k = 0.5 H

From the above concept,

M = 3 mH

Concept:

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 A carries current I₁ and coil B carries current I₂.

Due to current I₁, the flux produced is ϕ₁ which links with both the coils. Then the mutual inductance between two coils can be written as

Here, ϕ₁₂ is the part of the flux ϕ₁ linking with the coil 2

Calculation:

Number of turns = N₁ = N₂ = 500

Current (I) = 5 A

Flux produced in coil X (ϕ₁) = 10 mwb

Flux linked with Y (ϕ₁₂) = 80% of flux produced in coil 1 = 8 mwb