A 220V/12V single-phase transformer is designed for use in India and rated 100 VA at 50 Hz. Later, this unit is shipped to the USA where it is used as a 110V/6V transformer at 60 Hz. Which of the following statements is/are correct?

The correct option is: "For the same load current, the losses would be higher in the USA compared to that in India"

Let us analyze the transformer parameters and operation under the two different conditions to understand why this statement is correct.

1. Transformer Parameters in India:
Rated Voltage: $V_1=220\text{V}$ (primary) / $V_2=12\text{V}$ (secondary)
Rated Frequency: $f_{\text{India}}=50\text{Hz}$

2. Transformer Parameters in the USA:
Applied Voltage: $V_{\text{1,USA}}=110\text{V}$ (primary) / $V_{\text{2,USA}}=6\text{V}$ (secondary)
Operating Frequency: $f_{\text{USA}}=60\text{Hz}$

3. Magnetic Flux Density Analysis:
The induced EMF equation of a transformer is given by:

$E=4.44\cdot f\cdot N\cdot {\Phi }_m=4.44\cdot f\cdot N\cdot A\cdot B_m$

Here, $B_m$ is the peak magnetic flux density, which is proportional to the ratio of voltage to frequency ($V/f$). Let's compare this ratio for both cases:

In India:
${\left(\frac{V}{f}\right)}_{\text{India}}=\frac{220}{50}=4.4$

In the USA:
${\left(\frac{V}{f}\right)}_{\text{USA}}=\frac{110}{60}\approx 1.83$

Since ${\left(\frac{V}{f}\right)}_{\text{USA}}<{\left(\frac{V}{f}\right)}_{\text{India}}$, the peak magnetic flux density $B_m$ is significantly lower in the USA compared to India. This rules out the option stating peak magnetic flux density would be higher in the USA.

4. Core Loss Analysis (Hysteresis and Eddy Current Losses):
Let us check the components of core losses in both scenarios.
Hysteresis loss is given by:

$P_h=k_h\cdot f\cdot B_m^x$

Since $B_m\propto \frac{V}{f}$, we can write:

$P_h\propto f\cdot {\left(\frac{V}{f}\right)}^x\propto \frac{V^x}{f^{x-1}}$ (typically Steinmetz exponent $x\approx 1.6$)

Eddy current loss is given by:

$P_e=k_e\cdot f^2\cdot B_m^2\propto f^2\cdot {\left(\frac{V}{f}\right)}^2\propto V^2$

Since the operating voltage in the USA is half ($110\text{V}$ vs $220\text{V}$), both the core hysteresis and eddy current losses are significantly reduced in the USA compared to India.

5. Load Loss (Copper Loss) and Temperature Effect on Resistance:
For the same load current $I$, the nominal copper loss is given by $I^{2}R$. However, operating a transformer designed for a higher voltage rating (such as $220\text{V}$) at a lower terminal voltage ($110\text{V}$) to deliver the same load current means the power output is halved.
Additionally, the skin effect and eddy current losses in the conductors (winding eddy losses, which are part of stray load losses) are proportional to the square of the frequency ($f^2$).
Since the frequency in the USA ($60\text{Hz}$) is higher than in India ($50\text{Hz}$), the ac resistance of the windings increases due to the enhanced skin effect and proximity effect at $60\text{Hz}$. Thus, for the same load current, the winding losses (including stray load losses in the copper conductor) will be higher in the USA compared to India.