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Transformer coil temperature and power rating

Short answer

Reference resistance at 15°C is 11.852 Ω. With α=0.00125, the hot resistance yields core temperature ≈ 119.5°C. Power rating at 230 V = 230²/13.4 ≈ 3.95 kW.

A coil of stranded copper wire having a resistance of 12Ω at 25°C is embedded in the core of a large transformer supplied at 230 V. After the transformer has been in service for several hours, the resistance of the coil is found to be 13.4 Ω. What is the temperature of the core? Also find the power rating of the resistance. Assume temperature coefficient of wire as 0.00125/°C at 15°C.

The resistance of a copper coil rises with temperature, so the hot resistance tells us the core temperature.

Data

R₂₅ = 12 Ω, R_hot = 13.4 Ω, α₁₅ = 0.00125 per °C (as given), V = 230 V.

Step 1: Resistance at the reference temperature 15°C

R₂₅ = R₁₅ [1 + α₁₅ (25 − 15)] 12 = R₁₅ × (1 + 0.0125) = R₁₅ × 1.0125 R₁₅ = 11.852 Ω

Step 2: Core temperature

R_t = R₁₅ [1 + α₁₅ (t − 15)] 13.4 = 11.852 [1 + 0.00125 (t − 15)] 1 + 0.00125 (t − 15) = 1.1306 t − 15 = 104.5 t = 119.5°C

(Converting α to 25°C, α₂₅ = 0.00125/1.0125 = 0.001235, gives the same result.)

Step 3: Power rating of the coil

In service, P = V²/R = 230²/13.4 = 3947.8 W

(At the cold value of 12 Ω it would be 4408 W.)

Answer: Core temperature ≈ 119.5°C; power ≈ 3.95 kW

Note: the usual α for copper at 15°C is about 0.004 per °C; the value 0.00125 is used here exactly as given.

✓ Answer checked by Claude Opus 5.5 (high effort)

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