Solution:

Given, the transformer rating = 200 kVA
Efficiency at full load = 98%
Maximum efficiency occurs at three-quarters of full load.

To calculate the efficiency at half load, we need to follow the below steps.

Step 1: Calculation of Full Load Current

Full Load Current (I1) = Transformer rating (kVA) / Transformer voltage (kV)

The voltage rating of the transformer is not given in the problem statement. Hence, we need to assume a voltage rating for further calculations.

Assuming the transformer voltage rating as 11 kV.

Full Load Current (I1) = 200 kVA / 11 kV ≈ 18.18 A

Step 2: Calculation of Three-quarters Load Current

As the maximum efficiency occurs at three-quarters of full load, we need to calculate the three-quarters load current.

Three-quarters Load Current (I3/4) = Full Load Current (I1) * √(3/4)

I3/4 = 18.18 * √(3/4) ≈ 15.86 A

Step 3: Calculation of Half Load Current

Half Load Current (I1/2) = Full Load Current (I1) * √(1/2)

I1/2 = 18.18 * √(1/2) ≈ 12.86 A

Step 4: Calculation of Efficiency at Half Load

To calculate the efficiency at half load, we need to use the following formula.

Efficiency = Output Power / Input Power

At half load, the output power is given by,

Output Power = Transformer rating (kVA) * Power Factor * Load Current (I1/2)

Assuming the power factor as 0.8.

Output Power = 200 kVA * 0.8 * 12.86 A ≈ 2060.8 W

The input power can be calculated as follows:

Input Power = Output Power / Efficiency

Efficiency at three-quarters load = 98%

Efficiency at half load can be obtained by interpolation.

Let x be the efficiency at half load.

(98 - x) / (98 - 94) = (15.86 - 12.86) / (15.86 - 18.18)

Solving the above equation, we get,

x ≈ 97.38%

Input Power = 2060.8 W / 0.9738 ≈ 2118.1 W

Therefore, the efficiency at half load is given by,

Efficiency = Output Power / Input Power = 2060.8 W / 2118.1 W ≈ 97.37%

Hence, the efficiency at half load is approximately 97.37%.