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.
Square
Coil
Let us consider
Size of core = a
Diameter of
circumscribing circle = d
The diameter of
circumscribing circle is
d = √ ( a^{2}
+ a^{2} )
= √ 2 ( a )
Therefore the
side of square a = d / √ 2
Gross area of
core A_{gi} = a^{2} = ( d / √ 2 )^{2} = 0.5d^{2}
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 ) * 0.5d^{2}
= 0.45d^{2}
Therefore the
ratio of net core area to the area of the circumscribing circle = 0.45d^{2}
/ ( π d^{2 }/ 4 )
= 0.58
Similarly the
the ratio of gross core area to the area of the circumscribing circle = 0.5d^{2}
/ ( π d^{2 }/ 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
A_{gi} = ab + [ ( a – b ) / 2 * b ] + [ ( a – b ) / 2 * b ]
A_{gi} =
ab + ab – b^{2 }
A_{gi} =
2ab – b^{2 }….....( 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 )
A_{gi} =
2 ( d Cos θ ) ( d Sin θ ) – ( d Sin θ )^{2
}
= 2d^{2}
Sin θ Cos θ – d^{2} Sin^{2} θ
= d^{2}
( Sin 2θ – Sin^{2} θ )
Differentiate A_{gi}
with respect to θ and equating to zero to get maximum core area
d / dθ ( A_{gi
}) = 0
d / dθ ( d^{2}
Sin 2θ – d^{2 }Sin^{2} θ ) = 0
( 2 d^{2}
Cos 2θ – d^{2 }2 Sin θ Cos θ ) = 0
Therefore d^{2}
( Sin 2 θ ) = 2 d^{2 }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 )
A_{gi} =
2 ( 0.85d ) ( 0.53d ) – ( 0.53d )^{2}
= 0.618d^{2}
Stacking factor
= Net core area / Gross core area
Therefore net
core area = 0.9 * Net core area
= 0.9 * ( 0.618d^{2}
)
= 0.56 d^{2}
The ratio of net
core area to the area of circumscribing circle
= 0.56 d^{2 }/
( π d^{2 }/ 4 )
= 0.71
Similarly the
ratio of gross core area to the area of circumscribing circle
= 0.618d^{2 }/
( π d^{2 }/ 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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