Solution:

Given data:
Frequency, f1 = 50 Hz
Frequency, f2 = 45 Hz
Voltage, V2 = 0.9 × V1 (Rated voltage)
Hysteresis loss at f1 = Eddy current loss at f1
We need to find out the core losses at 45 Hz and 90% of rated voltage.

Core losses in transformer:

The core losses in a transformer can be classified into two types:
1. Hysteresis loss
2. Eddy current loss

Hysteresis loss:

Hysteresis loss is the energy loss that occurs due to the reversal of magnetization in the core material. This loss is directly proportional to the frequency of the supply voltage and the B-H curve of the core material. The hysteresis loss is given by the following formula:
Ph = K1 × f × Bmax^1.6 watts

where,
K1 = Steinmetz constant
f = Frequency of the supply voltage
Bmax = Maximum flux density in the core

Eddy current loss:

Eddy current loss is the energy loss that occurs due to the circulating currents induced in the core material. This loss is directly proportional to the square of the supply voltage, the frequency of the supply voltage, and the thickness of the core material. The eddy current loss is given by the following formula:
Pe = K2 × f × V^2 × t^2 watts

where,
K2 = Constant depending on the core material
f = Frequency of the supply voltage
V = Supply voltage
t = Thickness of the core material

Calculation:

Given that the transformer has equal hysteresis and eddy current losses at rated excitation. Therefore, the total core loss at rated frequency and voltage is:
P1 = Ph + Pe

At 45 Hz and 90% of the rated voltage, the flux density in the core will be:
B2 = (V2 / 4.44 × f2 × N) / A

where,
N = Number of turns in the winding
A = Cross-sectional area of the core

Since the transformer has equal hysteresis and eddy current losses at rated excitation, the flux density in the core at 45 Hz and 90% of the rated voltage will be the same as that at rated frequency and voltage. Therefore, the hysteresis loss and eddy current loss at 45 Hz and 90% of the rated voltage will be the same as that at rated frequency and voltage.

Therefore, the total core loss at 45 Hz and 90% of the rated voltage will be:
P2 = P1

Substituting the values of f1, f2, V2, and P1 in the above equation, we get:
Ph + Pe = K1 × f1 × Bmax^1.6 + K2 × f1 × V1^2 × t^2
Ph + Pe = K1 × f2 × Bmax^1.6 + K2 × f2 × V2^2 × t^2

Since the hysteresis loss and eddy current loss at 45 Hz and 90% of the rated voltage are the same as that at rated frequency and voltage, the flux density in the core at 45 Hz and 90% of the rated voltage will also be the same as that at