Coefficient of Coupling or Magnetic Coupling Coefficient

Consider two coils having self inductance L1 and L2 placed very close to each other. Let the number of turns of the two coils be N1 and N2 respectively. Let coil 1 carries current i1 and coil 2 carries current i2.
        Due to current i1, the flux produced is Φ1 which links with both the coils. Then from the previous knowledge mutual inductance between two coils can be written as
              M = N1 Φ21/i1                                                    ...............(14)
       where  Φ21 is the part of the flux Φ1 linking with coil 2. Hence we can write, Φ21 = k1 Φ1.
...           M = N1 ( k1 Φ1)/i1                                             .................(15)
       Similarly due to current i2, the flux produced is Φ2 which links with both the coils. Then the mutual inductance between two coils can be written as
                 M = N2 Φ21/i2                                                              .........(16)
       where  Φ21 is the part of the flux Φlinking with coil 1. Hence we can write Φ21 =  k2 Φ2.
..              M = N2 (k2 Φ2)/i2                                                   ..................(17)
      Multiplying equations (15) and (17),

But             N1Φ1/i1 = Self induced of coil 1 = L1
                   N2Φ2/i2 = Self induced of coil 2 = L2
...                M2 = k1k2L1L2
...                 M = √(k1k2) √(L1L2)
Let                k = √(k1k2) 
..                  M = k √(L1L2)                                                           ............(18)
      where k is called coefficient of coupling.
...                    k = M/(√(L1L2))                                                                .........(19)
      The coefficient of coupling gives idea about the magnetic coupling between the two coils. So when the entire flux in one coil links with the other, the coupling coefficient is maximum. The maximum value of k is unity. Thus when k = 1, the coupled coils are called tightly or perfectly coupled coils. Also the mutual inductance between the two coils is maximum with k =1. The maximum value of the mutual inductance is given by
                            M =  √(L1L2)                                                              ..............(20)
       When the two coils are at greater distance in space, the value of k is very small. Then the two coils are called loosely coupled coils.
Key Point : k is non-negative fraction and has maximum value of unity.
      For iron core coupled circuits : k = 0.99
      For air core coupled circuits : k = 0.4 to 0.7
       From equations (19) and (20), the coefficient of coupling can be alternatively defined as the ratio of the actual mutual inductance present between the two coils to the maximum possible value of the mutual inductance.
       The coefficient of coupling between the two coils can also be expressed interms of the reactance offered by the self inductance and mutual inductance as

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