Transformer Loss Analysis

This solution explains how changes in supply voltage and frequency affect the hysteresis and eddy current losses in a single-phase transformer. We are given that both the supply voltage and frequency are increased by 10%.

Hysteresis Loss Principles

Hysteresis loss ($P_h$) occurs due to the magnetic hysteresis effect in the transformer core. It represents the energy lost as the magnetic domains within the core material are repeatedly rearranged due to the alternating magnetic field.

The formula for hysteresis loss is generally given by:

$$ P_h \propto f \cdot B_{max}^{x} $$

where:

• $f$ is the supply frequency.
• $B_{max}$ is the maximum magnetic flux density in the core.
• $x$ is the Steinmetz coefficient (a material property, typically between 1.5 and 2.5).

Eddy Current Loss Principles

Eddy current loss ($P_e$) is caused by circulating currents (eddy currents) induced within the transformer core by the changing magnetic flux. These currents flow through the resistance of the core material, generating heat and causing energy loss.

The formula for eddy current loss is:

$$ P_e \propto f^2 \cdot B_{max}^{2} $$

where the symbols have the same meaning as above.

Flux Density Calculation

In a transformer, the induced voltage ($V$) is directly proportional to the rate of change of flux, and the maximum flux ($ \Phi_{max}$) is related to the supply voltage ($V$) and frequency ($f$) by the following relationship:

$$ V \propto f \cdot \Phi_{max} $$

The maximum magnetic flux density ($B_{max}$) is proportional to the maximum flux ($ \Phi_{max}$):

$$ B_{max} \propto \Phi_{max} $$

Combining these, we get:

$$ B_{max} \propto \frac{V}{f} $$

Now, let's consider the given changes:

• Original Voltage = $V$, Original Frequency = $f$.
• New Voltage $V' = V + 0.10V = 1.10V$.
• New Frequency $f' = f + 0.10f = 1.10f$.

Let's calculate the new maximum flux density ($B'_{max}$):

$$ B'_{max} \propto \frac{V'}{f'} = \frac{1.10V}{1.10f} = \frac{V}{f} $$

This shows that $B'_{max} = B_{max}$. The maximum flux density remains unchanged when both voltage and frequency are increased by the same percentage.

Hysteresis Loss Percentage Change

We know $P_h \propto f \cdot B_{max}^{x}$. Let the original loss be $P_{h,old}$ and the new loss be $P_{h,new}$.

$$ \frac{P_{h,new}}{P_{h,old}} = \frac{f' \cdot (B'_{max})^{x}}{f \cdot B_{max}^{x}} $$

Since $f' = 1.10f$ and $B'_{max} = B_{max}$, we substitute these values:

$$ \frac{P_{h,new}}{P_{h,old}} = \frac{(1.10f) \cdot (B_{max})^{x}}{f \cdot B_{max}^{x}} = 1.10 $$

So, $P_{h,new} = 1.10 \cdot P_{h,old}$.

The percentage change in hysteresis loss is:

$$ \% \Delta P_h = \frac{P_{h,new} - P_{h,old}}{P_{h,old}} \times 100 = \frac{1.10 P_{h,old} - P_{h,old}}{P_{h,old}} \times 100 $$

$$ \% \Delta P_h = \frac{0.10 P_{h,old}}{P_{h,old}} \times 100 = 10\% $$

Therefore, the hysteresis loss increases by 10%.

Eddy Current Loss Percentage Change

We know $P_e \propto f^2 \cdot B_{max}^{2}$. Let the original loss be $P_{e,old}$ and the new loss be $P_{e,new}$.

$$ \frac{P_{e,new}}{P_{e,old}} = \frac{(f')^2 \cdot (B'_{max})^{2}}{f^2 \cdot B_{max}^{2}} $$

Substituting $f' = 1.10f$ and $B'_{max} = B_{max}$:

$$ \frac{P_{e,new}}{P_{e,old}} = \frac{(1.10f)^2 \cdot (B_{max})^{2}}{f^2 \cdot B_{max}^{2}} = \frac{(1.10)^2 f^2 \cdot B_{max}^{2}}{f^2 \cdot B_{max}^{2}} $$

$$ \frac{P_{e,new}}{P_{e,old}} = (1.10)^2 = 1.21 $$

So, $P_{e,new} = 1.21 \cdot P_{e,old}$.

The percentage change in eddy current loss is:

$$ \% \Delta P_e = \frac{P_{e,new} - P_{e,old}}{P_{e,old}} \times 100 = \frac{1.21 P_{e,old} - P_{e,old}}{P_{e,old}} \times 100 $$

$$ \% \Delta P_e = \frac{0.21 P_{e,old}}{P_{e,old}} \times 100 = 21\% $$

Therefore, the eddy current loss increases by 21%.

Transformer Loss Summary

Based on the calculations, when the supply voltage and frequency are both increased by 10%:

• Hysteresis loss increases by 10%.
• Eddy current loss increases by 21%.

The percentage changes are 10% and 21%, respectively.