3 June 2019

Transformer : Advantages of Stepped Core Over Square Core

Core of Transformer

  • The cross section of the transformer core may be rectangular, square or stepped. Generally the circular coils are preferred for distribution transformer whereas the square and stepped cores are preferred for power transformer. 
  • The cross section of the core may be rectangular for shell type transformer. 
  • The rectangular coil is suitable for small and low voltage transformers.

Why the stepped core is used in place of square core in the large rating transformer?


  • The diameter of the circumscribing circle for square / rectangular coil is larger than the diameter of stepped core of same area of cross section therefore the length of mean turns of winding is reduced in the stepped core. 
  • Therefore the length of mean turns in the stepped core reduced which result in reduction of copper winding cost and copper losses.
  • However as the number of steps increases of different sizes, the cost of labour charges for shearing and assembling of different laminations are increases.

stepped-core-of-the-transformer


Square Coil
Let us consider
Size of core = a
Diameter of circumscribing circle = d
The diameter of circumscribing circle is
d = √ ( a2 + a2 )
= √ 2 ( a )
Therefore the side of square a = d / √ 2
Gross area of core Agi = a2 = ( d / √ 2 )2 = 0.5d2
Stacking factor = Net core area / Gross core area
Let the stacking factor is 0.9
Net core area = ( 0.9 ) * Gross core area
                       = ( 0.9 ) * 0.5d2
                       = 0.45d2
Therefore the ratio of net core area to the area of the circumscribing          circle = 0.45d2 / ( π d2 / 4 )
           = 0.58
Similarly the the ratio of gross core area to the area of the circumscribing circle = 0.5d2 / ( π d2 / 4 )
          = 0.64
Two Stepped Core
Let the length of rectangular = a
Breath of rectangular = b
Diameter of circumscribing circle = d
Angle between the length of rectangular and diagonal = θ
The two stepped core can be divided into three rectangular parts.
Gross core area Agi = ab + [ ( a – b ) / 2 * b ] + [ ( a – b ) / 2 * b ]
Agi = ab + ab – b2
Agi = 2ab – b….....( 1 )

Now Cos θ = a / d
a = d Cos θ ……….( 2 )
Sin θ = b / d
b = d Sin θ …….…( 3 )

Substitute value of a and b in the equation ( 1 )
Agi = 2 ( d Cos θ ) ( d Sin θ ) – ( d Sin θ )
      = 2d2 Sin θ Cos θ – d2 Sin2 θ
      = d2 ( Sin 2θ – Sin2 θ )
Differentiate Agi with respect to θ and equating to zero to get maximum core area
d / dθ ( Agi ) = 0
d / dθ ( d2 Sin 2θ – d2 Sin2 θ ) = 0
( 2 d2 Cos 2θ – d2 2 Sin θ Cos θ ) = 0
Therefore d2 ( Sin 2 θ ) = 2 d2 Cos 2θ
       ( Sin 2 θ / Cos 2θ ) = 2
                         Tan 2 θ = 2
                                2 θ = tan – 1 ( 2 )
                                   θ = [ 1 / 2 ] tan – 1 ( 2 ) = 31.72….( 4 )
Using equation ( 2 ), ( 3 ) and ( 4 )
a = d Cos θ
  = d Cos ( 31.72 )
  = 0.85d
b = d Sin θ
  = d Sin ( 31.72 )
  = 0.53d
Substitute value of a and b in the equation ( 1 )
Agi = 2 ( 0.85d ) ( 0.53d ) – ( 0.53d )2
      = 0.618d2
Stacking factor = Net core area / Gross core area
Therefore net core area = 0.9 * Net core area
= 0.9 * ( 0.618d2 )
= 0.56 d2
The ratio of net core area to the area of circumscribing circle
= 0.56 d2 / ( π d2 / 4 )
= 0.71
Similarly the ratio of gross core area to the area of circumscribing circle
= 0.618d2 / ( π d2 / 4 )
= 0.79
Finally we conclude that
Ratio of Core area to area of Circumscribing Circle
Square core
Two stepped core ( Cruciform core )
Net core area / Area of circumscribing circle
0.58
0.71
Gross core area / Area of circumscribing circle
0.64
0.79

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