Electrical Engineering ⇒ Topic : Mutual Inductance

MUTUAL INDUCTANCE

The induced voltage in coil 2 is due to a change in flux (Φ₁₂) and we can write it using Faraday's law

                                                                               .............. (1)

Here Φ₁₂ is related to i₁. So e₂ is proportional to the rate of change of i₁   i.e..

                                                                                          ................. (2)

In Eq. (2), M₁₂ is the constant of proportionality known as mutual inductance between two coils. It is to be noted that M is dimensionally equivalent to self inductance L and its unit is same as L, i.e., Henry (H).

From Eqs. (1) and (2), we can write

                                                                         .................. (3)

As mutual coupling is bilateral and a change in current i₂ in circuit 2 causes a mutually induced voltage in circuit 1. Again we can write

                                                                                .................... (4)

If the flux and current are linearly related, Eq. (3) and Eq. (4) can be given by

                                                                                     .................... (5)

If the permeability of the mutual flux path is assumed to be constant

                                      M₁₂ = M₂₁ = M

Equations (5)  suggest that the mutual inductance between two coils is defined as the weber turns in one coil due to current through the other coil

  Consider two coils A and B placed adjacent to each other as shown in Figure. If a current I₁ flows in the coil A, a flux is set up and a part Φ₁₂ (mutual flux) of this flux links the coil B. If current in coil A is varied, the mutual flux also varies and hence an e.m.f. is induced in the coil B. The e.m.f. induced in coil B is termed as mutually induced e.m.f. Note that coil B is not electrically connected to coil A ; the two coils being magnetically linked

                                                   FIGURE (A)

The larger the rate of change of cun-ent in coil A, the greater is the e.m.f. induced in coil B. In other words, mutually induced e.m.f. in coil B is directly proportional to the rate of change of current in coil A i. e.,Mutually induced e.m.f. in coil B cc Rate of change of current in coil A

                                                                                               (in magnitude)      ............  (1)

where M is a constant called mutual inductance between the two coils. The unit of mutual inductance is henry (H). If in expression. (1), e_m= 1 volt, dl/dt = 1 A/sec, then, M = 1 H.

Hence mutual inductance between two coils is 1 henry if current changing at the rate of 1 A/sec in one coil induces an e.m.f. of 1 V in the other coiL

     Mutual inductance comes into picture when two coils are placed close together in such a way that flux produced by one links the other. We say then that the two coils are coupled. Each coil has its own inductance but in addition, there is further inductance due to the induced voltage produced by coupling between the coils. We call this further inductance as mutual inductance. We say the two coils are coupled together by mutual inductance. The terms magnetic or inductive coupling are sometimes used.

Note. The magnitude of mutually induced e.m.f. in coil B (secondary) is e_M= M dl₁/dt where dl₁ is the change of current in coil A (primary). However, the magnitude and direction of mutually induced e.m.f. in coil B should be written as

   The minus sign is because the mutually induced e.m.f. sends current in coil B in such a direction so as to produce magnetic flux which opposes the change in flux produced by change in current in coil A. In fact, minus sign represents Lenz's law mathematically.