Self Inductance

What Is Self Inductance?

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 say that self-inductance is a phenomenon where there is the induction of a voltage in a current-carrying wire.

The self-induced emf present in the coil will resist the rise of current when the current increases, and it also resists the fall of current if the current decreases. In essence, the direction of the induced emf is opposite to the applied voltage if the current is increasing, and the direction of the induced emf is in the same direction as the applied voltage if the current is falling.

The above property of the coil exists only for the changing current, which is the alternating current and not for the direct or steady current. Self-inductance is always opposing the changing current and is measured in Henry (SI unit).

Induced current always opposes the change in current in the circuit, whether the change in the current is an increase or a decrease one. Self-inductance is a type of electromagnetic induction.

Self-inductance Formula

We can derive an expression for the self-inductance of a coil from Faraday’s law of electromagnetic induction.

V_L = −N (dϕ / dt)

Where:

V_L = induced voltage in volts

N = number of turns in the coil

dφ / dt = rate of change of magnetic flux in webers / second

Alternatively, the induced voltage in an inductor may also be expressed in terms of the inductance (in henries) and the rate of change of current.

V_L = −L (di / dt)

Or

E = −L (di / dt)

Self-inductance of a Solenoid

We will understand the concept with the help of an example. We will take a solenoid having N turns; let its length be ‘l’, and the area of the cross-section be ‘A’, where current I is flowing through it. There will be a magnetic field ‘B’ at any given point in the solenoid. Therefore, the magnetic flux per turn will be equal to B × area of each turn.

However, B = (μ₀NI)/l

Therefore, magnetic flux per turn = (μ₀NIA)/l

Now, the total magnetic flux (φ) that is connected with the solenoid will be given by the product of flux present through every turn and the total number of turns.

Φ = (μ₀NIA) x N /l

That is, Φ = (μ₀N²IA) /l ….(eq 1)

If L is the self-inductance of the solenoid, then

Φ = LI …..(eq 2)

Combining the equations (1) and (2) from above, we get

L = (μ₀N²A) /l

If you have a core that is made up of a magnetic material of permeability μ, then

L = (μN²A) /l

Also Read: Electromagnetic Induction

Mechanical Equivalent of Self-inductance

The self-induced emf is also called the back emf as it opposes any change in the current in a circuit. Physically, self-inductance acts as inertia. In mechanics, it is also the electromagnetic analogue of mass. So, work needs to be done against the back emf (V_L) in establishing the current. This work done is stored as magnetic potential energy and is given by:

W = (1 / 2) LI².

This is the energy required to build up a current I in the inductor.

This expression reminds us of mv ² /2 for the (mechanical) kinetic energy of a particle of mass m and shows that L is analogous to m (i.e., L is electrical inertia and opposes both the growth and decay of current in a circuit).

Mutual Inductance

Mutual inductance is the opposition to the change of current in one coil due to the presence of a second coil. If a medium of relative permeability μ_r is present, the mutual inductance would be M =μ_r μ₀ n₁n₂π r²₁ l, where n₁ and n₂ are the numbers of turns of the two coils per unit length, and r₁ is the radius of the inner coil and l is the length of the coil.

Read More: Mutual Inductance

Difference between Mutual Inductance and Self Inductance

Uses of Self Inductance

The major function of an inductor is to store electrical energy in the form of a magnetic field. Inductors are used in the following:

• Tuning circuits
• Sensors
• Store energy in a device
• Induction motors
• Transformers
• Filters
• Chokes
• Ferrite beads
• Inductors used as relays

Limitations of Inductors

1) An inductor is limited in its current-carrying capacity by its resistance and dissipates heat.

2) Inductors in pure form are difficult to manufacture due to stray effects and size, whereas capacitors are relatively easy to manufacture with negligible stray effects.

3) Inductors may affect the nearby components in the circuit with their magnetic fields.

Solved Questions

Question 1:

Self-inductance of an inductor is given by ________.

a) L = N⁄S

b) L = 1⁄S

c) L = N2⁄S

d) L = N2

Answer: c)

The self-inductance of an inductor is obtained by making use of the relation

L = N2⁄S

Where, N is the number of turns

S is the reluctance of the coil (A/Wb).

Question 2:

Self-inductance depends on ________.

a) Permeability

b) Permittivity

c) Plank’s constant

d) Rydberg constant

Answer: a)

The self-inductance of an inductor is obtained by making use of the relation

L = N2⁄S

Where, N is the number of turns

S is the reluctance of the coil (A/Wb)

As reluctance depends on permeability, the self-inductance of a coil depends on permeability.

Question 3:

The phase angle is always ________ for an RL circuit.

a) Positive

b) Negative

c) 0

d) 90

Answer: b)

For a series resistance and inductance circuit, the phase angle is always a negative value because the current will always lag the voltage.

Question 4:

What is φ in terms of voltage?

a) Φ = cos^-1V/V_R

b) Φ = cos^-1V*V_R

c) Φ = cos^-1V_R/V

d) Φ = tan^-1V/V_R

Answer: c)

From the voltage triangle, we get cos φ = V_R/V.

Hence, φ = cos^-1V_R/V.

Question 5:

What is sin ϕ from the impedance triangle?

a) XL/R

b) XL/Z

c) R/Z

d) Z/R

Answer: b)

In the impedance triangle, the base is R, the hypotenuse is Z, and the height is X_L.

So, sin ϕ = X_L/Z.