From Flux to EMF: Demystifying Mutual Inductance

Introduction

Magnetic flux and electromotive force might seem like abstract physics concepts — but they’re the key to understanding how coils interact, transformers work, and energy gets transferred across air gaps. In this post, we’ll take you on a clear, intuitive journey:

• Starting from magnetic flux fundamentals
• Showing how flux in one coil induces voltage in a neighboring coil
• Defining and explaining mutual inductance
• Demonstrating how this concept forms the backbone of transformers, inductors, and many electromagnetic systems

Whether you’re learning mutual inductance for the first time or brushing up on your electromagnetic theory, by the end you’ll have a clear grasp of how flux becomes EMF in coupled coils.

Mutual Inductance is the phenomenon where the magnetic field generated by current in one coil induces an EMF (voltage) in an adjacent coil. This is different from self-induction, where the changing magnetic field around a coil induces EMF within the same coil.

When two or more coils are magnetically linked by a common magnetic flux, they exhibit mutual inductance. The changing current in one coil induces EMF in the adjacent coil. This is the fundamental operating principle of transformers, motors, and generators.

Positioning Dependence:  The amount of mutual inductance is heavily influenced by the relative positioning of the two coils: Close positioning and alignment maximise magnetic flux linkage, resulting in a high mutual inductance. Greater distance or different angles reduce flux linkage, decreasing the induced EMF and mutual inductance.

Interference:  Mutual inductance can sometimes be undesirable, as stray or leakage inductance from one coil may interfere with nearby components. This effect can often be mitigated with electrical shielding to ground potential.

Mutual Inductance between Coils

The mutual inductance between two coils can be significantly increased by:

Positioning the coils on a common soft iron core, which enhances magnetic coupling by concentrating and directing the magnetic flux.

Increasing the number of turns in either coil, as more turns mean more flux linkage and thus a greater induced EMF. When the two coils are tightly wound on top of each other over a common soft iron core, unity coupling is said to exist.  In this configuration, flux leakage is minimised, allowing nearly perfect flux linkage between the coils. In such cases, the mutual inductance M  can be maximised, as virtually all the magnetic flux generated by the first coil links to the second coil. 

This principle of enhanced mutual inductance through close coupling and an iron core is foundational in transformers, where efficient energy transfer relies on maximising flux linkage between the primary and secondary windings.

M = (μ₀ x μ_r xN₁ x N₂ x A) / ℓ

Mutual Inductance

The mutual inductance M₁₂ between two coils, where coil one L₁ has a current I₁ and N₁ turns, and coil two L₂ has N₂  turns, is the result of the magnetic field generated by the current in coil one that links with coil two. 

The mutual inductance M₁₂  ( Henry, H) of coil two with respect to coil one depends on their relative positioning and is calculated by the formula:

M₁₂ = N₂  x Φ₁₂  / I ₁

where: N₂  is the number of turns in coil two, Φ₁₂  is the magnetic flux through coil two due to the current I₁ in coil one. ( Wb)

The mutual inductance depends on factors such as the distance and orientation of the coils and any shared magnetic core, which can increase flux linkage and therefore the mutual inductance value. This principle is key in transformers and other devices that rely on magnetic coupling between coils.

The mutual inductance between two coils is reciprocal, meaning that the mutual inductance of coil one with respect to coil two M₂₁ is the same as the mutual inductance of coil two with respect to coil one M₁₂. 

Thus, we can write: M₁₂ = M₂₁ = M

This relationship holds regardless of the size, number of turns, relative position, or orientation of the two coils. Mutual inductance M  represents the magnetic coupling between two inductors or coils, influenced by factors such as distance and arrangement.

In contrast: Self-inductance (L) characterises an inductor as an individual component, describing the induced EMF within itself due to its own changing current.

Mutual inductance reflects the interaction between two inductors, where a changing current in one coil induces an EMF in the other coil through magnetic coupling.

L₁ =( μ₀ x μ_r x N₁² x A) / ℓ and L₂ = (μ₀ x μ_r x N₂² x A ) / ℓ

By cross-multiplying the two equations above, the mutual inductance, M that exists between the two coils can be expressed in terms of the self inductance of each coil.giving us a final and more common expression for the mutual inductance between the two coils of:

M² = L₁L₂

Mutual Inductance Between Coils

In practice, 100% magnetic coupling between two coils (L₁ and L₂) is unattainable due to flux leakage and positional factors. Although perfect coupling is impossible, the magnetic coupling can approach 100% in specially designed inductive coils.

The fraction of the total magnetic flux that links both coils is represented by the coefficient of coupling, denoted by the letter k . This coefficient expresses how closely the coils are magnetically coupled and is defined as:

Thus, the mutual inductance between the two coils in real conditions is given by

M = k √(L₁L₂)

When the coefficient of coupling, k is equal to 1, (unity) such that all the lines of flux of one coil cuts all of the turns of the second coil, that is the two coils are tightly coupled together, the resulting mutual inductance will be equal to the geometric mean of the two individual inductances of the coils.

Also when the inductances of the two coils are the same and equal, L₁ is equal to L₂, the mutual inductance that exists between the two coils will equal the value of one single coil as the square root of two equal values is the same as one single value as shown.

M = √L₁L₂ = L

Example

The diagram shows a coil of N₂ turns and a radius of R₂ surrounding a long solenoid of length l₁, radius R₁ and N₁ turns.

a. What is the mutual inductance of the two coils?

b. If N₁ = 500 turns, N₂ = 10 turns, R₁ = 3.10cm, l₁=75cm and the current in the solenoid is changing at a rate of 200A/s. What is the emf induced in the surrounding coil?

Solution:

The magnetic flux ф₂₁ through the surrounding coil is 

  ф₂₁ = B₁ x ㄫ x R₁²

  = ( μ₀ x  N₁ x  I₁ x ㄫ x R₁² ) / ℓ

If we now use the equation for mutual inductance

   M = (N₂ x ф₂₁ ) / I₁

                       M = N₂ / I₁  x ( μ₀ x N₁ x I₁ x  ㄫ x R₁²) / ℓ₁

    M = ( μ₀ x N₁ x N₂ x  ㄫ x R₁² ) /  ℓ₁

substituting the values into the equation gives:

  M = ((4ㄫx10^-7) x 500 x 10 x ㄫ x (0.0310) ²  )/ 0.750

  M = 2.53×10^-5H

We can then calculate the emf form the following equation:

           Emf = -MdI₁/ dt

Emf = -(2.53×10^-5 ) 200

  Emf = -5.06×10^-3V

Example 2.

A solenoid of 500 turns is wound on an iron core of relative permeability 800. The length and radius of the solenoid are 40 cm and 3 cm respectively. Calculate the average emf induced in the solenoid if the current in it changes from 0 to 3 A in 0.4 second.

Solution

N = 500 turns; µ_r = 800 ; μ_o = 4πx10^-7, l = 40 cm = 0.4 m; r = 3 cm = 0.03 m;

di = 3 – 0 = 3 A; dt = 0.4 s

Area = πr² = 3.14 x ( 3×10^-2)² = 2.8×10^-3 m²

n = N/l = 500 / 0.4 = 1250

  L = μ_o x μ_r x n²  x A x l

  L = 4πx10^-7 x 800 x 500 x 1250² x 2.8×10^-3 x 0.4

  L = 1.77H

  Emf = Ldi/dt

  Emf = (1.77 x 3) / 0.4 

   Emf = 13.275V

Example 3:

Two inductors whose self-inductances are given as 75mH and 55mH respectively, are positioned next to each other on a common magnetic core so that 75% of the lines of flux from the first coil are cutting the second coil. Calculate the total mutual inductance that exists between the two coils

  M = k √L₁L₂

  M = 0.73  √75 x 55

  M = 48.2mH

Example 4:

When two coils having inductances of 5H and 4H respectively were wound uniformly onto a non-magnetic core, it was found that their mutual inductance was 1.5H. Calculate the coupling coefficient that exists between.

  k = M / √L₁L₂ = 1.5 / √5×4 = 0.335 = 33.5%

 

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