Transformer Loss Calculation Explained

Transformers are essential electrical devices used to transfer electrical energy between two or more circuits through electromagnetic induction. When a transformer operates, it incurs various energy losses. These losses are primarily categorized into two types: copper loss and iron loss (also known as core loss). Understanding how these losses behave under different load conditions, especially at half load, is crucial for assessing transformer efficiency.

Understanding Transformer Losses

• Copper Loss ($\text{P}_{\text{cu}}$): This loss occurs in the windings of the transformer due to the resistance of the copper conductors. It is also known as $\text{I}^2\text{R}$ loss. Copper loss is variable and depends on the load current flowing through the windings. Specifically, it is proportional to the square of the load current. As the load on the transformer changes, the current changes, and thus the copper loss changes. Mathematically, if 'x' represents the fraction of full load, then copper loss at any load 'x' is given by: $P_{cu,x} = x^2 \times P_{cu,full\_load}$
• Iron Loss ($\text{P}_{\text{i}}$): This loss occurs in the magnetic core of the transformer and consists of two components: hysteresis loss and eddy current loss. These losses depend primarily on the maximum flux density in the core and the frequency of the alternating current. Since transformers typically operate at a constant voltage and frequency, the magnetic flux density in the core remains almost constant regardless of the load. Therefore, iron loss is considered to be constant from no-load to full-load operation. It does not vary significantly with the change in load current.

Given Transformer Data

We are provided with the full load copper loss and iron loss for a transformer:

• Full load copper loss ($P_{cu,full\_load}$) = $\text{6400 W}$
• Full load iron loss ($P_{i,full\_load}$) = $\text{5000 W}$

Calculating Transformer Losses at Half Load

We need to determine the copper loss and iron loss when the transformer operates at half load. Half load means the load factor (x) is 0.5.

Copper Loss at Half Load

Since copper loss is proportional to the square of the load, at half load, the current is half, and thus the copper loss will be $(0.5)^2$ or $0.25$ times the full load copper loss.

Let $P_{cu,half\_load}$ be the copper loss at half load.

$P_{cu,half\_load} = (\text{Load fraction})^2 \times P_{cu,full\_load}$

$P_{cu,half\_load} = (0.5)^2 \times 6400 \text{ W}$

$P_{cu,half\_load} = 0.25 \times 6400 \text{ W}$

$P_{cu,half\_load} = 1600 \text{ W}$

Iron Loss at Half Load

As discussed, iron loss is constant regardless of the load on the transformer, assuming constant voltage and frequency. Therefore, the iron loss at half load will be the same as the full load iron loss.

Let $P_{i,half\_load}$ be the iron loss at half load.

$P_{i,half\_load} = P_{i,full\_load}$

$P_{i,half\_load} = 5000 \text{ W}$

Final Transformer Loss Results

Based on the calculations, the copper loss and iron loss at half load are:

• Copper Loss at half load = $\text{1600 W}$
• Iron Loss at half load = $\text{5000 W}$