Synchronous Generator Base Impedance Calculation

Understanding the base impedance is crucial in power system analysis, especially when working with per-unit systems. The base impedance provides a reference value against which all other impedances in the system are normalized. For a three-phase system like a synchronous generator, the base impedance can be calculated using the generator's base apparent power (MVA) and base voltage (kV).

Impedance Calculation Formula

The base impedance ($\text{Z}_{\text{base}}$) for a three-phase system is derived from the base voltage ($\text{V}_{\text{base}}$) and base apparent power ($\text{S}_{\text{base}}$) using the following formula:

$$\text{Z}_{\text{base}} = \frac{(\text{V}_{\text{base}})^2}{\text{S}_{\text{base}}}$$

Where:

• $\text{Z}_{\text{base}}$ is the base impedance in Ohms ($\Omega$).
• $\text{V}_{\text{base}}$ is the base line-to-line voltage in kilovolts (kV).
• $\text{S}_{\text{base}}$ is the three-phase base apparent power in megavolt-amperes (MVA).

Given Generator Parameters

From the question, we are provided with the following specifications for the synchronous generator:

• Rated Apparent Power ($\text{S}_{\text{base}}$) = 40 MVA
• Rated Line-to-Line Voltage ($\text{V}_{\text{base}}$) = 10 kV
• Frequency = 50 Hz (Note: The frequency is not directly used in the calculation of base impedance but is a standard parameter for synchronous machines.)

Step-by-Step Base Impedance Derivation

To find the base impedance of the synchronous generator, we substitute the given values into the formula:

$$\text{Z}_{\text{base}} = \frac{(\text{V}_{\text{base}})^2}{\text{S}_{\text{base}}}$$

Substitute $\text{V}_{\text{base}} = 10 \text{ kV}$ and $\text{S}_{\text{base}} = 40 \text{ MVA}$:

$$\text{Z}_{\text{base}} = \frac{(10 \text{ kV})^2}{40 \text{ MVA}}$$

First, calculate the square of the base voltage:

$$(10 \text{ kV})^2 = 100 \text{ (kV)}^2$$

Now, perform the division:

$$\text{Z}_{\text{base}} = \frac{100 \text{ (kV)}^2}{40 \text{ MVA}}$$

$$\text{Z}_{\text{base}} = 2.5 \text{ } \Omega$$

Therefore, the base impedance of the synchronous generator is 2.5 Ohms.

Option Analysis

Let's compare our calculated base impedance with the given options:

The calculated base impedance of 2.5 $\Omega$ matches Option 4.