Electrical Engineering ⇒ Topic : Coefficient of Coupling

   The coefficient of coupling (k) between two coils is defined as the fraction of magnetic flux produced by the current in one coil that links the other

   When the entire flux of one coil links the other, coefficient of coupling is 1 (i.e., 100%). If only half the flux set up in one coil links the other, then coefficient of coupling is 0.5 (or 50%). If two coils have self-inductances L₁ and L₂, then mutual inductance M between them is given by

   where k = coefficient of coupling. Clearly, the mutual inductance M between the coils will be maximum when k= 1. If flux of one coil does not at all link with the other coil, then k = 0. Under condition, mutual inductance (M) between the coils will be zero.

   Proof. Consider two magnetically coupled coils 1 and 2 having N₁ and N₂ turns respectively (See Figure). The current I₁ flowing in coil 1 produces a magnetic flux Φ₁ Suppose the coefficient of coupling between the two coils is k. It means that flux  kΦ₁ links with coil 2. Then, by definition

                                                                                                              .......  (1)

where M₁₂ represents mutual inductance of coil 1 to coil 2.

The current I₂ flowing in coil 2 will produce flux Φ₂. Since the coefficient of coupling between the coils is k, it means that flux KΦ₂ will link with coil 1. Then

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

where M₂₁ represents mutual inductance of coil 2 to coil 1.

                                                                    FIGURE (A)

Mutual inductance between the two coils is exactly the same i.e.,M₁₂= M₂₁ = M

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

Expression (3) gives the relation between the mutual inductance of the two coils and their self-inductances.The reader may note that mutual inductance between the two coils will be maximum when k= 1. Obviously, the maximum value of mutual inductance between the two coils is = √L₁L₂

Hence, coefficient of coupling can also be defined as the ratio of the actual mutual inductance (M) between the two coils to the maximum possible value (√L₁L₂)

When two coils are wound on a single fen-omagnetic core as shown in Figure (a), effectively all of the magnetic flux produced by one coil links with the other. The coils are then said to be tightly coupled. Another way to ensure tight coupling is shown in Figure(b) where each turn of the secondary winding is side by side with one turn of primary winding. Coils wound in this fashion are said to be bifilar and it is called bifilar winding

        FIGURE (A)                                  FIGURE (B)                               FIGURE  (C)

When the two coils are air-cored as shown in Figure (c), then only a fraction of magnetic flux produced by one coil may link with the other coil. The coils are then said to be loosely coupled.