If each branch of a Delta circuit has impedance √3 Z, then each branch of the equivalent Y circuit has impedance:  

Explanation:

Delta to Star (Y) Transformation

Definition: Delta to Star (or Y) transformation is a mathematical technique used in circuit analysis to simplify circuits with three-terminal networks. This conversion makes it easier to calculate the equivalent impedance and solve complex circuits by transforming a Delta (Δ) network into an equivalent Star (Y) network.

Problem Statement: If each branch of a Delta circuit has an impedance of √3Z, what is the impedance of each branch in the equivalent Star (Y) circuit?

Solution:

For a three-phase circuit, the relationship between the impedances of the Delta (Δ) and Star (Y) networks is given by the following formula:

Impedance of each branch in the Y-network:

Z_Y = (Z_Δ × Z_Δ) / (Z_Δ + Z_Δ + Z_Δ)

Here, Z_Δ is the impedance of each branch in the Delta network, and Z_Y is the impedance of each branch in the equivalent Star network.

Given:

• Z_Δ = √3Z

Substitute the value of Z_Δ into the formula:

Z_Y = (√3Z × √3Z) / (√3Z + √3Z + √3Z)

Simplify the numerator:

Z_Y = (3Z²) / (3√3Z)

Cancel out common terms:

Z_Y = Z / √3

Therefore, the impedance of each branch in the equivalent Star (Y) circuit is Z / √3.

Correct Option: Option 1: Z / √3

Additional Information

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

Option 2: 3Z

This option is incorrect. The impedance of each branch in the Star network cannot exceed the impedance of the Delta network branches. The formula for Delta to Star conversion clearly shows that the impedance in the Star network is always lower than that in the Delta network.

Option 3: 3√3Z

This option is also incorrect. It vastly overestimates the equivalent impedance of the Star network. The correct Star network impedance is Z / √3, which is much smaller than the impedance values provided in this option.

Option 4: Z / 3

This option is incorrect because it underestimates the equivalent impedance. While the Star network impedance is smaller than the Delta network impedance, it is not as small as Z / 3. The correct value is Z / √3.

Conclusion:

Understanding the relationship between Delta and Star impedances is crucial for circuit analysis. The impedance of each branch in a Star network is always lower than that of the corresponding Delta network. The correct answer to the problem is Z / √3, as derived above. This transformation simplifies circuit calculations and is commonly used in power systems and network analysis.