Per Unit Impedance Overview

In power system analysis, the per unit (Pu) system is a convenient and widely used method to express quantities such as voltage, current, impedance, and power as a fraction of a chosen base value. This system simplifies calculations, especially when dealing with transformers and different voltage levels, as it eliminates the need to refer quantities across transformer turns ratios. It also helps in standardizing equipment parameters and makes fault calculations easier.

Impedance Definition in Per Unit

The per unit impedance of a system component is defined as the ratio of the actual impedance of the component (in ohms) to a chosen base impedance. The formula for per unit impedance is given by:

• $$ \text{Impedance (Pu)} = \frac{\text{Actual Impedance (Ohms)}}{\text{Base Impedance (Ohms)}} $$

To calculate the base impedance, we need a chosen base power and a chosen base voltage. For a three-phase system, these base values are typically expressed as:

• Base Power (\(S_B\)) in MVA (MegaVolt-Amperes)
• Base Voltage (\(V_B\)) in kV (kiloVolts)

Base Impedance Derivation

The base impedance (\(Z_B\)) for a three-phase system can be derived from the base power and base voltage. We know that for a three-phase system, power \(S = \sqrt{3} \times V \times I\). Also, by Ohm's Law, \(V = I \times Z\), so \(I = V/Z\). Substituting \(I\) into the power equation:

• $$ S_B = \sqrt{3} \times V_B \times \frac{V_B}{\sqrt{3} \times Z_B} $$
• $$ S_B = \frac{V_B^2}{Z_B} $$

Rearranging this formula to solve for base impedance \(Z_B\):

• $$ Z_B (\Omega) = \frac{V_B^2}{S_B} $$

Here, if \(V_B\) is in Volts and \(S_B\) is in VA, \(Z_B\) will be in Ohms. However, in power system analysis, voltage is often in kilovolts (kV) and power in megavolt-amperes (MVA). To maintain consistency and obtain \(Z_B\) in Ohms when \(V_B\) is in kV and \(S_B\) is in MVA, we convert units:

• $$ V_B (\text{Volts}) = V_B (\text{kV}) \times 10^3 $$
• $$ S_B (\text{VA}) = S_B (\text{MVA}) \times 10^6 $$

Substituting these into the base impedance formula:

• $$ Z_B (\Omega) = \frac{(V_B (\text{kV}) \times 10^3)^2}{S_B (\text{MVA}) \times 10^6} $$
• $$ Z_B (\Omega) = \frac{V_B^2 (\text{kV})^2 \times 10^6}{S_B (\text{MVA}) \times 10^6} $$
• $$ Z_B (\Omega) = \frac{(kV)_B^2}{MVA_B} $$

This is the standard formula for base impedance when base voltage is in kV and base power is in MVA.

Per Unit Impedance Calculation

Now, substituting the derived base impedance formula back into the per unit impedance definition:

• $$ Z(\text{Pu}) = \frac{Z(\Omega)}{Z_B(\Omega)} $$
• $$ Z(\text{Pu}) = Z(\Omega) \div \frac{(kV)_B^2}{MVA_B} $$
• $$ Z(\text{Pu}) = Z(\Omega) \times \frac{MVA_B}{(kV)_B^2} $$

This formula accurately represents the per unit impedance for a three-phase system, converting the actual impedance in Ohms to its per unit value based on the chosen MVA base and kV base.

Therefore, the correct expression for per unit impedance Z (Pu) in a 3-phase system is:

• $$ \text{Z(Pu)} = \text{Z}(\Omega) \times \frac{MVA_B}{(kV)_B^2} $$