Calculating Surge Impedance

The surge impedance of a transmission line is a characteristic impedance that represents the ratio of the voltage wave to the current wave propagating along the line. It is determined by the physical properties of the line, specifically its inductance and capacitance per unit length.

Surge Impedance Formula

The surge impedance ($\text{Z}_\text{s}$) of a transmission line can be calculated using the inductance per unit length ($\text{L}$) and the capacitance per unit length ($\text{C}$) with the following formula:

$$ \text{Z}_\text{s} = \sqrt{\frac{\text{L}}{\text{C}}} $$

Given Values

From the question, we are given the following values for the three-phase line:

• Capacitance per unit length ($\text{C}$) = 0.01 µF/unit length
• Inductance per unit length ($\text{L}$) = 2.5 mH/unit length

Converting Units

To use these values in the formula, we need to convert them to standard units (Farads and Henrys):

• $\text{C} = 0.01 \text{ µF} = 0.01 \times 10^{-6} \text{ F} = 1 \times 10^{-8} \text{ F}$
• $\text{L} = 2.5 \text{ mH} = 2.5 \times 10^{-3} \text{ H}$

Calculating Surge Impedance

Now, substitute the converted values into the surge impedance formula:

$$ \text{Z}_\text{s} = \sqrt{\frac{\text{L}}{\text{C}}} $$

$$ \text{Z}_\text{s} = \sqrt{\frac{2.5 \times 10^{-3} \text{ H}}{1 \times 10^{-8} \text{ F}}} $$

$$ \text{Z}_\text{s} = \sqrt{2.5 \times 10^{-3} \times 10^{8}} \text{ } \Omega $$

$$ \text{Z}_\text{s} = \sqrt{2.5 \times 10^{5}} \text{ } \Omega $$

$$ \text{Z}_\text{s} = \sqrt{25 \times 10^{4}} \text{ } \Omega $$

$$ \text{Z}_\text{s} = \sqrt{25} \times \sqrt{10^{4}} \text{ } \Omega $$

$$ \text{Z}_\text{s} = 5 \times 10^{2} \text{ } \Omega $$

$$ \text{Z}_\text{s} = 500 \text{ } \Omega $$

Result

The calculated surge impedance of the transmission line is 500 $\Omega$.