Correct Answer - Option 2 : 73 W
Concept:
Hysteresis losses: These are due to the reversal of magnetization in the transformer core whenever it is subjected to alternating nature of magnetizing force.
\({W_h} = \eta B_{max}^xfv\)
\({B_{max}} \propto \frac{V}{f}\)
Where
x is the Steinmetz constant
Bm = maximum flux density
f = frequency of magnetization or supply frequency
v = volume of the core
At a constant V/f ratio, hysteresis losses are directly proportional to the frequency.
Wh ∝ f
Eddy current losses: Eddy current loss in the transformer is I2R loss present in the core due to the production of eddy current.
\({W_e} = K{f^2}B_m^2{t^2}V\)
\({B_{max}} \propto \frac{V}{f}\)
Where,
K - coefficient of eddy current. Its value depends upon the nature of magnetic material
Bm - Maximum value of flux density in Wb/m2
t - Thickness of lamination in meters
f - Frequency of reversal of the magnetic field in Hz
V - Volume of magnetic material in m3
At a constant V/f ratio, eddy current losses are directly proportional to the square of the frequency.
We ∝ f2
Iron losses or core losses or constant losses are the sum of both hysteresis and eddy current losses.
Wi = Wh + We
At constant V/f ratio, Wi = Af + Bf2
Calculation:
The table below shows the given data.
| |
Frequency (f)
|
Core losses (W)
|
|
Case 1
|
40 Hz
|
50
|
|
Case 2
|
60 Hz
|
100
|
|
Case 3
|
50 Hz
|
Required value
|
The V/f ratio is constant in all the cases as shown in the above table.
Now, the equations for Case 1 and Case 2 are given below
Case 1: 50 = A (40) + B (40)2
Case 2: 100 = A (60) + B (60)2
By solving the above two equations,
A = 0.4166, B = 0.0208
Now, the required value for the Case 3 is
W3 = 0.4166 (50) + 0.0208 (50)2 = 72. 83 W
