Mutual inductance

MUTUAL INDUCTANCE

Consider the circuit shown in fig. 5.26, the changing current i, produces a variable flux 1 in the first coil. For the purpose of analysis, ϕ₁ is divided into two components.

ϕ₁ = ϕ₁₁ + ϕ₁₂ ... (6)

Here ϕ₁ is the total flux established by i₁, ϕ₁₁ = a part of ϕ₁. It links with coil 1 only but not with coil 2.

ϕ₁₂ = it is a part of 1. It links with both coils 2 and 1.

As, the flux linking with coil 2 changes, an e.m.f. is induced in the coil ϕ₂ and is given by

e₂ = N₂ dϕ₁₂ / dt … (7)

Also, e₂ is proportional to time rate of change of i. It is because ϕ₁₂ is produced by i,, therefore,

e₂ = M di₁/dt ... (8)

From equations (7) and (8), we can write that,

M = N₂ dϕ₁₂ / di₁ ... (9)

If the permeability is constant, the above equation becomes

M = N₂ dϕ₁₂ / i₁ ... (10)

Suppose that the second coil is connected to a voltage source. Let i₂ be the current flow and ϕ₂ be the total flux.

ϕ₂ = ϕ₂₂ + ϕ₂₁

Here, ϕ₂₁ is a part of ϕ₂ and links with coil 1. Then, the e.m.f. in the coil 1 = e₁.

In equations (10 & 13) M is called mutual inductance.

Definition of M

The mutual inductance between 2 coils is defined as the weber turns in one coil per ampere current in other coil. It is measured in henrys.

The mutual inductance is also defined as the ability of one coil to produce e.m.f. in other coil by induction when the current in the first changes.

Coefficient of coupling (K) or coefficient of magnetic coupling (K_M).

Consider the fig. 5.19, the fraction of the total flux produced by coil 1 linking coil 2 is ϕ₁₂ / ϕ₁ . It is called coefficient of coupling. Thus

K = ϕ₁₂ / ϕ₁ … (14)

Also K = ϕ₂₁ / ϕ₂  ... (15)

Multiplying equations (10) & (13), we get

From the above expression, we can say that

M / L₁, and M / L₂ are in geometric progression.

Note:

 (i) The values of K depends on spacing, orientation of the coils and on the permeability of the medium.

(ii) It is a non-negative number and is independent of the reference directions of the currents. d lio

(iii) If the coils are at great distance apart, M is very small and hence K.

(iv) For iron-core coupled circuits, K may be as high as 0.99.

(v) For air-core coupled circuits, K varies between 0.4 and 0.8.

(vi) The maximum value of K is 1. Hence M_max = √L₁ L₂.