In the case where the voltage bases are the same, the new per unit impedance is obtained from the formula:

Explanation:

Per Unit Impedance Calculation

Definition: Per unit impedance is a normalized value used in power system analysis, which simplifies calculations and comparisons. It is expressed as a fraction of the base impedance of the system. When the voltage bases remain the same, the new per unit impedance can be calculated using a specific formula that adjusts the value based on the change in system base powers.

Correct Formula:

The correct formula for calculating the new per unit impedance when the voltage bases are the same is:

Option 1: \(\rm Z_{p u}^{\text {new }}=\frac{Z_{p u}^{\text {old }} S_{B}^{\text {new }}}{S_{B}^{\text {old }}}\)

This formula is derived based on the relationship between the old and new base quantities in the power system. It ensures that the impedance values are properly scaled with respect to the change in the base power values.

Explanation:

The per unit system is a convenient way of representing electrical quantities such as impedance, voltage, and current relative to a set of base values. In the case where the voltage bases remain constant, only the base power values change. The per unit impedance is scaled proportionally to the ratio of the new base power (\(S_{B}^{\text{new}}\)) to the old base power (\(S_{B}^{\text{old}}\)). This adjustment ensures consistency in calculations and comparisons across different systems.

Let’s break down the formula:

• \(Z_{p u}^{\text{old }}\): The per unit impedance value based on the old base quantities.
• \(S_{B}^{\text{new }}\): The new base power value for the system.
• \(S_{B}^{\text{old }}\): The old base power value for the system.
• \(Z_{p u}^{\text{new }}\): The per unit impedance value based on the new base quantities.

The formula effectively scales the old per unit impedance by the ratio of the new base power to the old base power. This is essential for maintaining consistency in system analysis when the base power values are updated.

Advantages of Using Per Unit System:

• Simplifies power system calculations by normalizing values.
• Facilitates comparison between different system components.
• Eliminates the need for unit conversions during calculations.
• Provides a clear representation of electrical quantities relative to the system base values.

Important Information

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

Option 2: \(\rm Z_{p u}^{\text {old }}=\frac{Z_{p u}^{\text {new }} S_{B}^{\text {new }}}{S_{B}^{\text {old }}}\)

This formula is incorrect because it reverses the relationship between the old and new per unit impedance values. Instead of scaling the old impedance to obtain the new impedance, it attempts to derive the old impedance from the new impedance. While mathematically possible, this is not the standard approach used in power system analysis.

Option 3: \(\rm Z_{p u}^{\text {new }}=\frac{Z_{p u}^{\text {old }} S_{B}^{\text {old }}}{S_{B}^{\text {new }}}\)

This formula is incorrect as it inversely scales the old per unit impedance with respect to the ratio of the base power values. It would result in an incorrect representation of the new per unit impedance, leading to errors in system analysis.

Option 4: \(\rm Z_{p u}^{\text {new }}=\frac{Z_{p u}^{\text {old }} S_{B}^{\text {new }} V_{B}^{\text {old }}}{S_{B}^{\text {old }} V_{B}^{\text {new }}}\)

This formula introduces voltage base values (\(V_{B}^{\text{old}}\) and \(V_{B}^{\text{new}}\)) into the calculation, which is unnecessary when the voltage bases are the same. When the voltage bases do not change, these terms cancel out, and the formula simplifies to Option 1. Including voltage base values in this case complicates the calculation unnecessarily.

Conclusion:

The correct formula for calculating the new per unit impedance when the voltage bases remain constant is Option 1. This formula ensures proper scaling of the old per unit impedance value based on the ratio of the new base power to the old base power. It is essential for consistent and accurate power system analysis, especially when transitioning between different base quantities.