A large transformer operating at no load draws an exciting current Io of 5A, when the primary is connected to a 120 V, 60 Hz source. From a wattmeter test, it is known that iron losses are equal to 180 W. Calculate the reactive power absorbed by the core. 

Explanation:

Reactive Power Absorbed by the Core

Problem Statement: A large transformer operating at no load draws an exciting current \(I_o\) of 5A when the primary is connected to a 120 V, 60 Hz source. The iron losses (core losses) are given as 180 W. We are required to calculate the reactive power absorbed by the core.

Solution:

To determine the reactive power absorbed by the core, we will use the following concepts:

• The total exciting current (\(I_o\)) drawn by the transformer at no load can be resolved into two components:
  • Active component (\(I_w\)): This component is responsible for the iron losses (core losses).
  • Reactive component (\(I_m\)): This component is responsible for producing the magnetic flux in the transformer core.
• Iron losses (\(P_{iron}\)) are related to the active component (\(I_w\)) of the exciting current.
• Reactive power (\(Q\)) is calculated using the reactive component (\(I_m\)) and the applied voltage (\(V\)).

Step-by-Step Calculation:

Step 1: Calculate the Active Component of the Exciting Current (\(I_w\))

The active component (\(I_w\)) is responsible for the iron losses (\(P_{iron}\)) in the transformer and can be calculated using the formula:

\[ P_{iron} = V \cdot I_w \]

Rearranging the formula to solve for \(I_w\):

\[ I_w = \frac{P_{iron}}{V} \]

Substitute the given values:

• \(P_{iron} = 180 \, \text{W}\)
• \(V = 120 \, \text{V}\)

\[ I_w = \frac{180}{120} = 1.5 \, \text{A} \]

The active component of the exciting current is \(I_w = 1.5 \, \text{A}\).

Step 2: Calculate the Reactive Component of the Exciting Current (\(I_m\))

The total exciting current (\(I_o\)) is the vector sum of the active component (\(I_w\)) and the reactive component (\(I_m\)). Thus, we can use the Pythagorean theorem to calculate \(I_m\):

\[ I_o^2 = I_w^2 + I_m^2 \]

Rearranging the formula to solve for \(I_m\):

\[ I_m = \sqrt{I_o^2 - I_w^2} \]

Substitute the given and calculated values:

• \(I_o = 5 \, \text{A}\)
• \(I_w = 1.5 \, \text{A}\)

\[ I_m = \sqrt{5^2 - 1.5^2} = \sqrt{25 - 2.25} = \sqrt{22.75} \approx 4.77 \, \text{A} \]

The reactive component of the exciting current is \(I_m \approx 4.77 \, \text{A}\).

Step 3: Calculate the Reactive Power (\(Q\))

The reactive power (\(Q\)) absorbed by the core can be calculated using the formula:

\[ Q = V \cdot I_m \]

Substitute the given and calculated values:

• \(V = 120 \, \text{V}\)
• \(I_m \approx 4.77 \, \text{A}\)

\[ Q = 120 \cdot 4.77 \approx 572.4 \, \text{var} \]

The reactive power absorbed by the core is approximately \(572 \, \text{var}\).

Final Answer: The reactive power absorbed by the core is 572 var, which corresponds to Option 2.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: 600 var

This value is incorrect. While it might seem plausible as a rounded-up value, the actual calculation yields \(572 \, \text{var}\). Hence, this option is not accurate.

Option 3: 180 var

This option is incorrect because \(180 \, \text{W}\) represents the iron losses (active power), not the reactive power. Reactive power is calculated based on the reactive component (\(I_m\)) of the exciting current, as shown in the solution.

Option 4: 360 var

This value is incorrect as it does not match the actual calculation. The reactive power is directly proportional to the reactive current component and the applied voltage, and the correct value is \(572 \, \text{var}\).

Conclusion:

The reactive power absorbed by the core of the transformer is approximately \(572 \, \text{var}\), which corresponds to Option 2. This result is derived based on the active and reactive components of the exciting current and the applied voltage. Understanding the relationship between these components is crucial for analyzing transformer performance at no load.