Inductance – Definition, Derivation, Types, Examples

Magnetism has a mystical quality about it. Its capacity to change metals like iron, cobalt, and nickel when touched piques children’s interest. Repulsion and attraction between the magnetic poles by observing the shape of the magnetic field created by the iron filling surrounding the bar magnet will be learned. According to physicists, the forces that govern both magnetism and electricity are substantially greater than gravity in electromagnetism. The maglev train, which is suspended above its tracks, is a wonderful demonstration of immense power.

Inductance

Inductance is an electrical circuit attribute that opposes any change in current in the circuit. Electrical circuits have an intrinsic feature called inductance. Whether desired or not, it will always be found in an electrical circuit. The inductance of a straight wire carrying electricity with no iron element in the circuit will be lower. Because the inductance of an electrical circuit opposes any change in current in the circuit, it is equivalent to inertia in mechanics.

Magnetic flux that is proportional to the rate of change of the magnetic field is known as induction. The induced EMF across a coil is related to the rate at which the current through it changes. Inductance is the proportionality constant in that relationship. H is the SI unit for inductance (henry). It is denoted by the letter L. The amount of inductance required to produce an EMF of 1 (V) volt in a coil when the current change rate is 1 Henry is defined as 1 H (Henry).

Factors affecting Inductance

The following are some of the factors that influence inductance:

1. The inductor’s wire has a specific number of turns.
2. The material that was used to make the core.
3. The core’s appearance.

Faraday established the Electromagnetic Induction Law, which states that by altering the magnetic flux, an electromotive force is induced in the circuit. The concept of induction is derived from Faraday’s law of electromagnetic induction. The electromotive force generated to counteract a change in current at a specific time interval is known as inductance.

Derivation of Inductance

Take a look at a DC source that has the switch turned on. When the switch is turned on, the current flows from zero to a specific value, causing a change in the flow rate. Consider the flux shift caused by current flow. The flux change is measured in terms of time, as follows:

dφ/dt

Use Faraday’s law of electromagnetic induction to solve the problem.

E = N(dϕ/dt)

Where, N is the coil’s number of turns, and E is the induced EMF across the coil.

Write the above equation as follows using Lenz’s law:

E = -N(dϕ/dt)

For computing the value of inductance, the previous equation is adjusted.

E = -N(dϕ/dt)

∴ E = -L(di/dt)

N = dΦ = L di

NΦ = Li

Therefore,

Li = NΦ = NBA

Where, B denotes the flux density and A denotes the coil area.

Hl = Ni

Where H denotes the magnetic flux’s magnetizing force.

B = μH

Li = NBA

L = NBA/i = N2BA/Ni

N²BA/Hl = N²μHA/Hl

L = μN²A/l = μN²πr²/l

Types of Inductance

There are two types of inductance. They are self-induction and mutual induction. Let’s learn about them in more detail with proper definitions,

• Self Induction 

The magnetic flux associated with a coil or circuit changes anytime the electric current running through it changes. As a result, an emf is induced in the coil or circuit, which opposes the change that creates it, according to Faraday’s laws of electromagnetic induction. This phenomenon is known as ‘self-induction,’ and the induced emf is referred to as back emf, while the current created in the coil is referred to as induced current.

• Coefficient of self-induction: The current is proportional to the number of flux linkages with the coil, i.e., Nϕ is directly proportional, or Nϕ = Li (N is the number of turns in coil and Nϕ – total flux linkage). Hence The coefficient of self-induction is L = (Nϕ/i).
• If i = 1amp, N = 1 then, L = ϕ i.e. When the current in a coil is 1 amp, the coefficient of self-induction is equal to the flux associated with the coil.
• Faraday’s second law induced emf e = -N(dϕ/dt). Which gives e = -L(di/dt); If  di/dt = amp/sec then |e| = L. When the rate of change of current in the coil is unity, the coefficient of self induction is equal to the emf induced in the coil.
• Units and dimensional formula of ‘L’ : It’s S.I. unit, weber/Amp = (Tesla × m²)/ Amp = (N × m)/Amp² = Joule/Amp² = (Coulomb × volt)/Amp² = (volt × sec)/amp = (ohm × sec).

But practical unit is henry (H). It’s dimensional formula [L] = [ML²T^-2A^-2]

• Dependence of self-inductance (L): ‘L’ is determined by the number of turns (N), the area of cross section (A), and the permeability of the medium, not by the current flowing or changing, but by the number of turns (N), the area of cross-section (A), and the permeability of the medium (μ). ‘L’ does not play a role in the circuit until there is a steady current running through it. Only when there is a change in current does ‘L’ enter the picture.
• The magnetic potential energy of inductor: In order to create a continuous current in the circuit, the source emf must work against the coil’s self-inductance, and any energy expanded for this work is stored in the coil’s magnetic field, which is referred to as magnetic potential energy (U).

U = 1/2 (Li)i = Nϕi/2

The various formulae for L

1. Circular coil, L = μ₀πN²r/2
2. Solenoid, L = μ₀N²r/l = μ₀n²Al
3. Toroid, L = μ₀N²r/2
4. Square coil, L = 2√2μ₀N²a/π

• Mutual Induction 

When the current going through a coil or circuit varies, so does the magnetic flux coupled to a neighboring coil or circuit. As a result, an emf will be induced in the next coil or circuit. Mutual induction is the term for this occurrence.

• Coefficient of mutual induction: N₂ϕ₂ is the total flux linked with the secondary due to current in the primary, and N₂ϕ₂ is directly proportional to i₁ = N₂ϕ₂ = Mi₁, where N₁ is the number of turns in the primary, N₂ is the number of turns in the secondary, ϕ₂ is the flux linked with each turn of the secondary, i₁ is the current flowing through the primary, and M is the mutual inductance coefficient.
• According to Faraday’s second law emf induces in secondary e₂=-N(dϕ₂/dt); e₂=-M(di₁/dt)
• If di₁/dt = 1Amp/sec then |e₂|=M. When the rate of change of current in the main coil is unity, the mutual induction coefficient is equal to the emf induced in the secondary coil.
• Units and dimensional formula of ‘M’: It’s S.I. unit, weber/Amp = (Tesla × m²)/ Amp = (N × m)/Amp² = Joule/Amp² = (Coulomb × volt)/Amp² = (volt × sec)/amp = (ohm × sec).

But practical unit is henry (H). It’s dimensional formula [M] = [ML²T^-2A^-2].

• Dependence of mutual inductance:

1. Both coils have the same number of turns (N₁, N₂).
2. Both coils’ self-inductance coefficients (L₁, L₂).
3. Coils cross-sectional area.
4. The nature of the material on which two coils are coiled or the magnetic permeability of the medium between the coils (μ_r).
5. Two coils are separated by this distance. (As d grows larger, M shrinks.)
6. Orientation of main and secondary coils (no flux relation M=0 for 90 degree orientation).
7. Between the primary and secondary coils, there is a ‘K’ coupling factor.

• Relation between M, L₁, and L₂: For two magnetically coupled coils M = K √L₁L₂, where k – coefficient of coupling or coupling factor which is defined as,

K = Magnetic flux linked in secondary / Magnetic flux linked in primary

0 ≤ K ≤ 1

The various formulae for M

1. Two concentric coplanar circular coils, M = πμ₀N₁N₂r²/2R
2. Two Solenoids, M = μ₀N₁N₂A/l
3. Two concentric coplanar square coils, M = μ₀2√2N₁N₂l²/πL

Combination of Inductance 

• series

If two mutually inducing self-inductance coils L₁ and L₂ are connected in series and separated by a large enough distance that mutual induction between them is insignificant, then net self-inductance L_s = L₁ + L₂.

When they’re near together, the net inductance is L_s = L₁ + L₂ ± 2M.

• Parallel 

When two mutually inducting self-inductance coils L₁ and L₂ are linked in parallel and separated by a large distance, the net inductance L is 1/L_p = 1/L₁ + 1/L₂.

∴ L_p = L₁L₂/L₁ + L₂

When they are in close proximity to one another,

L_p = L₁L₂ – M²/L₁ + L₂ ± 2M

Self Vs Mutual Inductance

Self Induction	Mutual Induction
The coil’s self-inductance is a property of the coil.	The characteristic of a pair of coils is mutual inductance.
When the main current in the coil declines, the induced current resists the decay of current in the coil.	When the main current in the coil declines, the induced current created in the nearby coil opposes the decay of the current in the coil.
When the coil’s primary current grows, the induced current opposes the expansion of current in the coil.	When the coil’s primary current grows, the induced current created in the adjoining coil opposes the coil’s current development.

Things to Keep in Mind

• Magnetic flux that is proportional to the rate of change of the magnetic field is known as induction.
• The induced EMF across a coil is related to the rate at which the current through it changes.
• Inductance is the proportionality constant in that relationship.
• H is the SI unit for inductance (henry). It is denoted by the letter L.
• Faraday’s law of electromagnetic induction is an electromagnetism fundamental law that describes how a magnetic field interacts with an electric circuit to produce an electromotive force (EMF).
• Faraday’s law is the name for this phenomena, which is known as electromagnetic induction.
• Mutual inductance is a property of a pair of conductors, whereas Self Inductance is a property of the conductors individually.

Sample Problems

Question 1: Three coils are wired together in a series. Each coil has an inductance of 5H, 4H, and 6H, respectively. Calculate the inductance equivalent.

Solution:

Given: L₁ = 5H, L₂ = 4H, L₃ = 6H

The series inductance all sum as

L = L₁ + L₂ + L₃

∴ L = 5 + 4 + 6

∴ L = 15H

Question 2: What factors have an impact on inductance?

Answer:

The following are some of the factors that influence inductance:

1. The inductor’s wire has a specific number of turns.
2. The material that was used to make the core.
3. The core’s appearance.

Faraday established the Electromagnetic Induction Law, which states that by altering the magnetic flux, an electromotive force is induced in the circuit. The concept of induction is derived from Faraday’s law of electromagnetic induction. The electromotive force generated to counteract a change in current at a specific time interval is known as inductance.

Question 3: Define a coil’s self-inductance. Establish a S.I. unit for it.

Answer:

The property of a coil that opposes the growth or decay of the current flowing through it is known as self-induction.

Henry is the SI unit of self-inductance (H).

Question 4: Consider a 500-turn solenoid coiled on an iron core with a relative permeability of 800. The solenoid’s length is 40 cm, and its radius is 3 cm. The current changes from 0 to 3 A. Calculate the average induced emf for this change in current at 0.4 second intervals.

Solution:

N = 500 turns, μr = 800, Length = 40 cm = 0.4 m

Radius, r = 3 cm = 0.03 m

Change in current, di = 3 – 0 = 3 A

Change in time, dt = 0.4 sec

Self-inductance is given as

L = μN²Al = μ₀μ_rN²πr²/l

∴ L = (4)(3.14)(10^-7)(800)(500²)(3.14)(3 × 10^-2)²/0.4

∴ L = 1.77 H

ε = L di/dt = 1.77 × 3/0.4

∴ ε = 13.275 V

Question 5: Explain Combination of Inductance.

Answer:

• Series:

If two mutually inducting self inductance coils L₁ and L₂ are connected in series and separated by a large enough distance that mutual induction between them is insignificant, then net self inductance L_s = L₁ + L₂.

When they’re near together, the net inductance is L_s = L₁ + L₂ ± 2M.

• Parallel:

When two mutually inducting self-inductance coils L₁ and L₂ are linked in parallel and separated by a large distance, the net inductance L is 1/L_p = 1/L₁ + 1/L₂.

∴ L_p = L₁L₂/L₁+L₂

When they are in close proximity to one another,

L_p = L₁L₂ – M²/L₁ + L₂ ± 2M

Question 6: Write the difference between Self Inductance and Mutual Inductance.

Answer:

Self Induction	Mutual Induction
The coil’s self inductance is a property of the coil.	The characteristic of a pair of coils is mutual inductance.
When the main current in the coil declines, the induced current resists the decay of current in the coil.	When the main current in the coil declines, the induced current created in the nearby coil opposes the decay of the current in the coil.
When the coil’s primary current grows, the induced current opposes the expansion of current in the coil.	When the coil’s primary current grows, the induced current created in the adjoining coil opposes the coil’s current development.

Question 7: The inductance of a coil is 6 H, and the supply frequency is 70 Hz. What is the reactance?

Solution:

Given: L = 6H, f = 70Hz

Solution:

X = 2πfL

X = 2 × 3.14 × 70 × 6

X = 2637.6 Ω