Understanding Cable Leakage Resistance

Leakage resistance in a cable refers to the resistance offered by the insulating material to the flow of current from the conductor to the sheath or ground. This leakage current flows radially through the insulation along the entire length of the cable.

Imagine the insulation as a resistive path distributed along the cable's length. If you have a longer cable, you essentially have more parallel paths for the leakage current to flow. When resistances are in parallel, the total resistance decreases.

Therefore, the leakage resistance of a cable is inversely proportional to its length. This means that as the length of the cable increases, the leakage resistance decreases, and vice versa.

We can express this relationship mathematically as:

$$ R_L \propto \frac{1}{L} $$

Or, equivalently:

$$ R_L \times L = \text{Constant} $$

Calculating Leakage Resistance for 100 km Cable

We are given the following information:

• Leakage resistance for a 50 km cable ($R_{L1}$): 1 M$\Omega$
• Length of the first cable ($L_1$): 50 km
• Length of the second cable ($L_2$): 100 km
• We need to find the leakage resistance for the 100 km cable ($R_{L2}$).

Using the relationship $R_L \times L = \text{Constant}$, we can write:

$$ R_{L1} \times L_1 = R_{L2} \times L_2 $$

Now, we can substitute the given values into the equation:

$$ (1 \text{ M}\Omega) \times (50 \text{ km}) = R_{L2} \times (100 \text{ km}) $$

To find $R_{L2}$, we rearrange the equation:

$$ R_{L2} = \frac{(1 \text{ M}\Omega) \times (50 \text{ km})}{100 \text{ km}} $$

$$ R_{L2} = \frac{50 \text{ M}\Omega \cdot \text{km}}{100 \text{ km}} $$

The 'km' units cancel out:

$$ R_{L2} = \frac{50}{100} \text{ M}\Omega $$

$$ R_{L2} = 0.5 \text{ M}\Omega $$

So, the leakage resistance for a 100 km long cable is 0.5 M$\Omega$. This confirms our understanding that increasing the cable length decreases the leakage resistance.

Revision Table: Key Concepts

Additional Information: Cable Parameters

Cables used for transmission or distribution of electrical energy have several electrical parameters distributed along their length. These parameters are often represented using equivalent circuit models. For a simple model, these parameters include:

• Series Resistance (R): The resistance of the conductor material. This is directly proportional to the length ($R \propto L$).
• Series Inductance (L): The inductance due to the magnetic field around the conductor. This is also directly proportional to the length ($L_{inductance} \propto L$).
• Shunt Capacitance (C): The capacitance between the conductor and the sheath/ground, with the insulation as the dielectric. This is directly proportional to the length ($C \propto L$).
• Shunt Conductance (G) or Leakage Conductance: The conductance of the insulation, which is the reciprocal of leakage resistance. This represents the leakage current through the insulation. As leakage resistance is inversely proportional to length, leakage conductance is directly proportional to length ($G_L \propto L$).

Understanding these distributed parameters is crucial for analyzing the performance of long transmission cables, especially at high frequencies.