Question

A single phase transformer is rated at 40 kVA. The transformer has full-load copper losses of 800 W and iron losses of 500 W. Determine the transformer efficiency at half full-load and 0.8 power factor.

• A. 92%
• B. 90%
• C. 95.81%
• D. 89.31%

Hint:

Copper losses depend on load current, while iron losses remain constant. Efficiency calculations must account for both.

Answer 1

The Correct Option is C

Step 1: Determine the output power at half full-load.
Rated apparent power of the transformer is
\[ S = 40 \text{ kVA} \]
At half full-load,
\[ S_{\text{half-load}} = 0.5 \times 40 = 20 \text{ kVA} \]
Given power factor is 0.8, therefore output power is
\[ P_{\text{out}} = 20 \times 0.8 = 16 \text{ kW} \]
Step 2: Calculate iron losses.
Iron losses are constant and independent of load.
\[ P_{\text{iron}} = 500 \text{ W} \]
Step 3: Calculate copper losses at half full-load.
Copper losses vary as the square of the load.
\[ P_{\text{cu, half-load}} = (0.5)^2 \times 800 = 200 \text{ W} \]
Step 4: Calculate total losses at half full-load.
\[ P_{\text{loss}} = 500 + 200 = 700 \text{ W} \]
Step 5: Calculate transformer efficiency.
\[ \eta = \frac{P_{\text{out}}}{P_{\text{out}} + P_{\text{loss}}} \]
\[ \eta = \frac{16}{16 + 0.7} = 0.9581 \]
\[ \eta = 95.81% \]
Step 6: Conclusion.
The efficiency of the transformer at half full-load and 0.8 power factor is
\[ \boxed{95.81%} \]

Answer 2

Instead of building efficiency up from output power, we can work it out from the total loss fraction and check that fraction against each option.

At half load, output power \(= 0.5 \times 40 \text{ kVA} \times 0.8 = 16 \text{ kW}\).

Iron loss (constant) \(= 500\) W. Copper loss scales with the square of the load fraction: \((0.5)^2 \times 800 = 200\) W. Total loss \(= 700\) W \(= 0.7\) kW.

Input power \(= \) output \(+\) losses \(= 16 + 0.7 = 16.7\) kW.

Loss fraction of input: \(\dfrac{0.7}{16.7} \approx 0.0419\), i.e. about \(4.19\%\) of the input is lost.

1. 92%: Implies a loss fraction of \(8\%\), more than double the \(4.19\%\) actually lost at this load, too pessimistic for this transformer.
2. 90%: Implies a \(10\%\) loss fraction, even further from the computed \(4.19\%\).
3. 95.81%: Corresponds to a loss fraction of exactly \(100\% - 95.81\% = 4.19\%\), matching the computed loss fraction precisely.
4. 89.31%: Implies a loss fraction of about \(10.69\%\), far higher than the small losses that actually apply here.

Only the \(95.81\%\) option matches the loss fraction computed directly from the transformer's iron and copper losses at half load.

Therefore, the correct answer is 95.81%.