Converting Per Unit Inertia and Reactance Between MVA Bases

This problem requires converting the per unit (pu) values of inertia and reactance for an electrical machine from its original base rating (300 MVA) to a new common base rating (50 MVA). Understanding how these values change with the base power is crucial in power system analysis.

Given Machine Parameters

We are provided with the following information for the machine:

• Original Base Power ($S_{base1}$): 300 MVA
• Per Unit Inertia on Original Base ($M_1$): 10 pu
• Per Unit Reactance on Original Base ($X_1$): 4 pu
• New Common Base Power ($S_{base2}$): 50 MVA

Principles of Per Unit Base Conversion

Per unit values simplify calculations in power systems by normalizing quantities relative to specified base values (usually power and voltage). When the base power is changed, the pu values of certain parameters must be adjusted:

• Inertia Conversion (M): The per unit inertia constant (M) is generally defined in relation to the stored kinetic energy and the system base power. Specifically, if M is defined such that $M = \frac{H}{S_{base}}$ (where H is the inertia constant in seconds), the physical inertia H remains constant. Thus, the relationship between inertia on two different bases ($M_1$, $M_2$) and their respective base powers ($S_{base1}$, $S_{base2}$) is: $M_1 \times S_{base1} = M_2 \times S_{base2}$. This leads to the conversion formula $M_2 = M_1 \times \frac{S_{base1}}{S_{base2}}$.
• Reactance Conversion (X): Per unit reactance ($X_{pu}$) is calculated as the actual reactance ($X_{ohm}$) divided by the base impedance ($Z_{base}$). Since $Z_{base} = \frac{V_{base}^2}{S_{base}}$, the per unit reactance is $X_{pu} = \frac{X_{ohm} \times S_{base}}{V_{base}^2}$. Assuming the base voltage ($V_{base}$) remains constant during the base change, the per unit reactance is directly proportional to the base power. The conversion formula is $X_2 = X_1 \times \frac{S_{base2}}{S_{base1}}$.

Step-by-Step Calculation for Inertia

Using the inertia conversion formula:

$$ M_2 = M_1 \times \frac{S_{base1}}{S_{base2}} $$

Substitute the given values:

$$ M_2 = 10 \, \text{pu} \times \frac{300 \, \text{MVA}}{50 \, \text{MVA}} $$

$$ M_2 = 10 \, \text{pu} \times 6 $$

$$ M_2 = 60 \, \text{pu} $$

Step-by-Step Calculation for Reactance

Using the reactance conversion formula:

$$ X_2 = X_1 \times \frac{S_{base2}}{S_{base1}} $$

Substitute the given values:

$$ X_2 = 4 \, \text{pu} \times \frac{50 \, \text{MVA}}{300 \, \text{MVA}} $$

$$ X_2 = 4 \, \text{pu} \times \frac{1}{6} $$

$$ X_2 = \frac{4}{6} \, \text{pu} = \frac{2}{3} \, \text{pu} $$

Calculating the decimal value:

$$ X_2 \approx 0.67 \, \text{pu} $$

Final Calculated Values

The per unit values on the new 50 MVA common base are:

• Inertia (M₂): 60 pu
• Reactance (X₂): 0.67 pu

Therefore, the correct pair of values is (60, 0.67).