A single-phase 100 kVA, 1000 V / 100 V, 50 Hz transformer has a voltage drop of 5% across its series impedance at full load. Of this, 3% is due to resistance. The percentage regulation of the transformer at full load with 0.8 lagging power factor is

Understanding Transformer Voltage Regulation Calculation

This problem asks us to calculate the percentage voltage regulation for a single-phase transformer operating at full load with a lagging power factor. Voltage regulation quantifies how much the secondary terminal voltage changes from its no-load value to its full-load value, expressed as a percentage relative to the full-load voltage.

Transformer Specifications and Load Conditions

We are given the following details:

- Transformer Rating: 100 kVA
- Voltage Rating: 1000 V / 100 V
- Frequency: 50 Hz
- Total Series Impedance Voltage Drop (at full load): 5%
- Voltage Drop due to Resistance (at full load): 3%
- Load Condition: Full Load
- Power Factor (PF): 0.8 lagging

Determining the Percentage Reactance Drop

The total voltage drop across the transformer's series impedance (often denoted as $\%IZ$) is the vector sum of the voltage drop due to resistance ($\%IR$) and the voltage drop due to reactance ($\%IX$). The relationship between these percentage drops is given by the Pythagorean theorem:

$$(\%IZ{)}^2=(\%IR{)}^2+(\%IX{)}^2$$

From the problem statement:

- $\%IZ=5\%$
- $\%IR=3\%$

We can substitute these values into the equation to find the percentage reactance drop ($\%IX$):

$$(5\%{)}^2=(3\%{)}^2+(\%IX{)}^2$$

$$25=9+(\%IX{)}^2$$

Rearranging the equation to solve for $(\%IX{)}^2$:

$$(\%IX{)}^2=25-9$$

$$(\%IX{)}^2=16$$

Taking the square root of both sides gives the percentage reactance drop:

$$\%IX=\sqrt{16}=4\%$$

Therefore, the voltage drop due to reactance at full load is 4%.

Calculating Voltage Regulation

The approximate formula to calculate the percentage voltage regulation ($\text{VR}$) for a transformer at a lagging power factor is:

$$\text{VR}(\%)\approx \%IR\cos \varphi +\%IX\sin \varphi$$

We have the following values:

- $\%IR=3\%$
- $\%IX=4\%$
- Power Factor $\cos \varphi =0.8$

First, we need to find the value of $\sin \varphi$. Since the power factor is 0.8 lagging, the angle $\varphi$ is:

$$\varphi =\arccos (0.8)$$

$$\varphi \approx 36.87^{\circ }$$

Now, we find the sine of this angle:

$$\sin \varphi =\sin (36.87^{\circ })\approx 0.6$$

Substitute the values of $\%IR$, $\%IX$, $\cos \varphi$, and $\sin \varphi$ into the voltage regulation formula:

$$\text{VR}(\%)\approx (3\%)\times (0.8)+(4\%)\times (0.6)$$

$$\text{VR}(\%)\approx 2.4\%+2.4\%$$

$$\text{VR}(\%)\approx 4.8\%$$

Final Result

The calculated percentage voltage regulation for the transformer at full load with a 0.8 lagging power factor is 4.8%.