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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  1. Unless an unusually large ferromagnetizable mass is magnetically coupled to the core shared among the coils extending it outside the core of these coils? By "unusually large", I mean to suggest several hundreds of pounds or more. This may not have any effect on the behavior of the coils of this core, but may have an effect on any sets of coils with standard core sizes connected to this set of coils?

  2. "Tesla had combined the liquified air process, with his pneumatic piston-driven oscillator, even prior to the fire and Linde’s patent application. In addition, he had calculated that, for every 200 pounds of iron connected to the device, a full horsepower was added to it. Can you imagine how many horsepower were added by the connection of the oscillator and liquifaction device to the body of a submarine? If hypothetically a Type XXI German submarine were retro-fitted with this oscillator, and its displacement were 2,700 tons, that would produce an extra 27,000 h.p., only a small part of which would be needed to drive the oscillator, which involved very little friction and force to produce a large current."